Codes, differentially $\delta$-uniform functions and $t$-designs

TL;DR

This paper develops a new theoretical framework linking linear codes, $t$-designs, and differentially $oldsymbol{ extit{ extdelta}}$-uniform functions, leading to the construction of new codes and combinatorial designs beyond classical criteria.

Contribution

It introduces a generalized theory for punctured and shortened codes, extends the Assmus-Mattson theorem, and connects $2$-designs with differentially $ extdelta$-uniform functions, enabling new code and design constructions.

Findings

Constructed binary codes with new parameters and known weight distributions.

Produced new 2-designs and Steiner systems S(2, 4, 2^n).

Extended the theoretical understanding of codes supporting $t$-designs.

Abstract

Special functions, coding theory and $t$-designs have close connections and interesting interplay. A standard approach to constructing $t$-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$-designs. However, some linear codes hold $t$-designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$-transitive or $t$-homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$-designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$-designs is established, a generalized Assmus-Mattson theorem is developed, and a link between $2$-designs and differentially $\delta$-uniform functions and $2$-designs is built. With these general results, binary codes with new parameters and known weight distributions are obtained, new $2$-designs and Steiner system $S(2, 4, 2^n)$ are produced in this paper.

Tables

Table 1. Table 1 : The weight distribution of the code 𝒞 ​ ( D f ) 𝒞 subscript 𝐷 𝑓 \mathcal{C}(D_{f}) of Theorem 3.7

Weight	Multiplicity
$0$	$1$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}$	$2^n-1$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}$	$2^n-1$
${\nu }_f$	$1$

Table 2. Table 2 : The weight distribution of the shortened code 𝒞 ​ ( D f ) { t 1 } 𝒞 subscript subscript 𝐷 𝑓 subscript 𝑡 1 \mathcal{C}(D_{f})_{\{t_{1}\}} of Theorem 3.8

Weight	Multiplicity
$0$	$1$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}$	$\frac{{\nu }_f+2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}$	$\frac{{\nu }_f-2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$

Table 3. Table 3 : The weight distribution of the shortened code 𝒞 ​ ( D f ) { t 1 , t 2 } 𝒞 subscript subscript 𝐷 𝑓 subscript 𝑡 1 subscript 𝑡 2 \mathcal{C}(D_{f})_{\{t_{1},t_{2}\}} of Theorem 3.10

Weight	Multiplicity
$0$	$1$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}$	$\frac{\left({\nu }_f+2^{\frac{n-2}{2}}\right)\left({\nu }_f+2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}$	$\frac{\left({\nu }_f-2^{\frac{n-2}{2}}\right)\left({\nu }_f-2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$

Table 4. Table 4 : The weight distribution of the punctured code 𝒞 ​ ( D f ) { t 1 } 𝒞 superscript subscript 𝐷 𝑓 subscript 𝑡 1 \mathcal{C}(D_{f})^{\{t_{1}\}} of Theorem 3.12

Weight	Multiplicity
$0$	$1$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}-1$	$\frac{{\nu }_f-2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}$	$\frac{{\nu }_f+2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}-1$	$\frac{{\nu }_f+2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}$	$\frac{{\nu }_f-2^{\frac{n-2}{2}}}{2{\nu }_f}\left(2^n-1\right)$
${\nu }_f-1$	$1$

Table 5. Table 5 : The weight distribution of the punctured code 𝒞 ​ ( D f ) { t 1 , t 2 } 𝒞 superscript subscript 𝐷 𝑓 subscript 𝑡 1 subscript 𝑡 2 \mathcal{C}(D_{f})^{\{t_{1},t_{2}\}} of Theorem 3.14

Weight	Multiplicity
$0$	$1$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}-2$	$\frac{\left({\nu }_f-2^{\frac{n-2}{2}}\right)\left({\nu }_f-2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}-1$	$\frac{{\nu }_f^2-2^{n-2}}{2{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}-2^{\frac{n-4}{2}}$	$\frac{\left({\nu }_f+2^{\frac{n-2}{2}}\right)\left({\nu }_f+2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}-2$	$\frac{\left({\nu }_f+2^{\frac{n-2}{2}}\right)\left({\nu }_f+2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}-1$	$\frac{{\nu }_f^2-2^{n-2}}{2{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
$\frac{{\nu }_f}{2}+2^{\frac{n-4}{2}}$	$\frac{\left({\nu }_f-2^{\frac{n-2}{2}}\right)\left({\nu }_f-2^{\frac{n-2}{2}}-2\right)}{4{\nu }_f\left({\nu }_f-1\right)}\left(2^n-1\right)$
${\nu }_f-2$	$1$

Table 6. Table 6 : The weight distribution of the code 𝒞 ​ ( F ) 𝒞 𝐹 \mathcal{C}(F) of Theorem 3.17

Weight	Multiplicity
$0$	$1$
$2^{2m-1}-2^{m-1}$	$\left(2^l-1\right)2^{2m}$
$2^{2m-1}$	$2\left(2^{2m}-1\right)$
$2^{2m-1}+2^{m-1}$	$\left(2^l-1\right)2^{2m}$
$2^{2m}$	$1$

Table 7. Table 7 : The weight distribution of the code 𝒞 ​ ( F ) { t 1 } 𝒞 subscript 𝐹 subscript 𝑡 1 \mathcal{C}(F)_{\{t_{1}\}} of Theorem 3.18

Weight	Multiplicity
$0$	$1$
$2^{2m-1}-2^{m-1}$	$\left(2^l-1\right)\left(2^{2m-1}+2^{m-1}\right)$
$2^{2m-1}$	$2^{2m}-1$
$2^{2m-1}+2^{m-1}$	$\left(2^l-1\right)\left(2^{2m-1}-2^{m-1}\right)$

Table 8. Table 8 : The weight distribution of the code 𝒞 ​ ( F ) { t 1 , t 2 } 𝒞 subscript 𝐹 subscript 𝑡 1 subscript 𝑡 2 \mathcal{C}(F)_{\{t_{1},t_{2}\}} of Theorem 3.20

Weight	Multiplicity
$0$	$1$
$2^{2m-1}-2^{m-1}$	$\left(2^l-1\right)2^{m-2}\left(2^m+2\right)$
$2^{2m-1}$	$2^{2m-1}-1$
$2^{2m-1}+2^{m-1}$	$\left(2^l-1\right)2^{m-2}\left(2^m-2\right)$

Table 9. Table 9 : The weight distribution of the code 𝒞 ​ ( F ) { t 1 } 𝒞 superscript 𝐹 subscript 𝑡 1 \mathcal{C}(F)^{\{t_{1}\}} of Theorem 3.22

Weight	Multiplicity
$0$	$1$
$2^{2m-1}-2^{m-1}-1$	$\left(2^l-1\right)\left(2^{2m-1}-2^{m-1}\right)$
$2^{2m-1}-2^{m-1}$	$\left(2^l-1\right)\left(2^{2m-1}+2^{m-1}\right)$
$2^{2m-1}-1$	$2^{2m}-1$
$2^{2m-1}$	$2^{2m}-1$
$2^{2m-1}+2^{m-1}-1$	$\left(2^l-1\right)\left(2^{2m-1}+2^{m-1}\right)$
$2^{2m-1}+2^{m-1}$	$\left(2^l-1\right)\left(2^{2m-1}-2^{m-1}\right)$
$2^m-1$	$1$

Table 10. Table 10 : The weight distribution of the code 𝒞 ​ ( F ) { t 1 , t 2 } 𝒞 superscript 𝐹 subscript 𝑡 1 subscript 𝑡 2 \mathcal{C}(F)^{\{t_{1},t_{2}\}} of Theorem 3.24

Weight	Multiplicity
$0$	$1$
$2^{2m-1}-2^{m-1}-2$	$2^{m-2}\left(2^l-1\right)\left(2^m-2\right)$
$2^{2m-1}-2^{m-1}-1$	$2^{2m-1}\left(2^l-1\right)$
$2^{2m-1}-2^{m-1}$	$\left(2^l-1\right)2^{m-2}\left(2^m+2\right)$
$2^{2m-1}-2$	$2^{2m-1}-1$
$2^{2m-1}-1$	$2^{2m}$
$2^{2m-1}$	$2^{2m-1}-1$
$2^{2m-1}+2^{m-1}-2$	$2^{m-2}\left(2^l-1\right)\left(2^m+2\right)$
$2^{2m-1}+2^{m-1}-1$	$2^{2m-1}\left(2^l-1\right)$
$2^{2m-1}+2^{m-1}$	$\left(2^l-1\right)2^{m-2}\left(2^m-2\right)$
$2^m-2$	$1$

Equations

$$\left(\genfrac{}{}{0ex}{}{\nu }{t}\right)\lambda =\left(\genfrac{}{}{0ex}{}{k}{t}\right)b.$$

$$\mathrm{Supp}(\mathbf{c})=\{i:c_i\ne 0,i\in \mathcal{P}(\mathcal{C})\}.$$

$$\sum _{i=0}^{\nu }{i}^t{A}_i=\sum _{i=0}^t(-1{)}^i{A}_i^{\perp }\left[\sum _{j=i}^{t}j!S(t,j)q^{m-j}(q-1{)}^{j-i}(\genfrac{}{}{0ex}{}{\nu -i}{\nu -j})\right],$$

$$w-\lfloor \frac{w+q-2}{q-1}\rfloor <d.$$

$$w^{\perp }-\lfloor \frac{w^{\perp }+q-2}{q-1}\rfloor <d^{\perp }.$$

$${\lambda }_{T_1}^{T_0}=\lambda (t_0,t_1),$$

$$F(x+a)+F(x)=b,$$

$$\underset{a\in \mathrm{GF}(2^n{)}^{\ast },b\in \mathrm{GF}(2^m)}{\max }\delta (a,b)=\delta ,$$

$$\{\delta (a,b):a\in \mathrm{GF}(2^n{)}^{\ast },b\in \mathrm{GF}(2^m)\}=\{0,2^s\}.$$

$${\mathcal{W}}_F(\lambda ,\mu )=\sum _{x\in \mathrm{GF}(2^n)}(-1{)}^{{\mathrm{Tr}}_{2^n/2}\left(\lambda F(x)+\mu x\right)},$$

$$A_k({\mathcal{C}}^T)=\sum _{i=0}^t\frac{\left(\genfrac{}{}{0ex}{}{\nu -t}{k}\right)\left(\genfrac{}{}{0ex}{}{k+i}{t}\right)\left(\genfrac{}{}{0ex}{}{t}{i}\right)}{\left(\genfrac{}{}{0ex}{}{\nu -t}{k-t+i}\right)\left(\genfrac{}{}{0ex}{}{\nu }{t}\right)}{A}_{k+i}(\mathcal{C}).$$

$${\pi }^T:\mathcal{C}$$

$$(c_i{)}_{i\in \mathcal{P}(\mathcal{C})}$$

$$A_k({\mathcal{C}}^T)=\sum _{t_1=0}^t\sum _{T_1\subseteq T,\#T_1=t_1}{\mu }_{T_1}({\mathcal{W}}_{k+t_1}(\mathcal{C})),$$

$${\mu }_{T_1}({\mathcal{W}}_{k+t_1}(\mathcal{C}))=$$

$$=$$

$$A_k({\mathcal{C}}^T)=\sum _{t_1=0}^t\left(\genfrac{}{}{0ex}{}{t}{t_1}\right)\frac{\left(\genfrac{}{}{0ex}{}{\nu -t}{k}\right)\left(\genfrac{}{}{0ex}{}{k+t_1}{t}\right)}{\left(\genfrac{}{}{0ex}{}{\nu -t}{k-t+t_1}\right)\left(\genfrac{}{}{0ex}{}{\nu }{t}\right)}{A}_{k+t_1}(\mathcal{C}).$$

$$A_k({\mathcal{C}}_T)=\frac{\left(\genfrac{}{}{0ex}{}{k}{t}\right)\left(\genfrac{}{}{0ex}{}{\nu -t}{k}\right)}{\left(\genfrac{}{}{0ex}{}{\nu }{t}\right)\left(\genfrac{}{}{0ex}{}{\nu -t}{k-t}\right)}{A}_k(\mathcal{C}).$$

$${\pi }_T:\mathcal{C}(T)$$

$$(c_i{)}_{i\in \mathcal{P}(\mathcal{C})}$$

$$A_k({\mathcal{C}}_T)={\mu }^T({\mathcal{W}}_k(\mathcal{C})),$$

$${\mu }^T({\mathcal{W}}_k(\mathcal{C}))=$$

$$=$$

$$A_k({\mathcal{C}}^T)=\sum _{i=0}^t\frac{\left(\genfrac{}{}{0ex}{}{\nu -t}{k}\right)\left(\genfrac{}{}{0ex}{}{k+i}{t}\right)\left(\genfrac{}{}{0ex}{}{t}{i}\right)}{\left(\genfrac{}{}{0ex}{}{\nu -t}{k-t+i}\right)\left(\genfrac{}{}{0ex}{}{\nu }{t}\right)}{A}_{k+i}(\mathcal{C}).$$

$$\mathcal{C}(D_f)=\{\left({\mathrm{Tr}}_{2^n/2}(x{d}_0)+y,\dots ,{\mathrm{Tr}}_{2^n/2}(x{d}_{{\nu }_f-1})+y\right):x\in \mathrm{GF}(2^n),y\in \mathrm{GF}(2)\}.$$

$$A_k(\mathcal{C}(D_f{)}_{\{t_1\}})=\frac{{\nu }_f-k}{{\nu }_f}{A}_k\left(\mathcal{C}(D_f)\right).$$

$$A_k(\mathcal{C}(D_f{)}_{\{t_1,t_2\}})=\frac{({\nu }_f-k)({\nu }_f-k-1)}{{\nu }_f({\nu }_f-1)}{A}_k\left(\mathcal{C}(D_f)\right).$$

$$A_k(\mathcal{C}(D_f{)}^{\{t_1\}})=\frac{{\nu }_f-k}{\nu }{A}_k\left(\mathcal{C}(D_f)\right),$$

$$A_{k-1}(\mathcal{C}(D_f{)}^{\{t_1\}})=\frac{k}{{\nu }_f}{A}_k\left(\mathcal{C}(D_f)\right).$$

$$A_k(\mathcal{C}(D_f{)}^{\{t_1,t_2\}})=A_k\left(\mathcal{C}(D_f{)}_{\{t_1,t_2\}}\right),$$

Full text

{AMS}

94B05, 05B05, 06E30.

1 Introduction

We start with a brief recall of $t$-designs. Let $\mathcal{P}$ be a set of $\nu$ elements and $\mathcal{B}$ a multiset of $b$ $k$-subsets of $\mathcal{P}$, where $\nu\geq 1$, $b\geq 0$ and $1\leq k\leq\nu$. Let $t$ be a positive integer satisfying $1\leq t\leq\nu$. The pair $\mathbb{D}=(\mathcal{P},\mathcal{B})$ is called a $t$-$(\nu,k,\lambda)$ design, or simply $t$-design, if every $t$-subset of $\mathcal{P}$ is contained in exactly $\lambda$ elements of $\mathcal{B}$. The elements of $\mathcal{P}$ are called points, and those of $\mathcal{B}$ are referred to as blocks.

When $\mathcal{B}=\emptyset$, i.e., $b=0$, we put $\lambda=0$ and call $(\mathcal{P},\emptyset)$ a $t$-$(\nu,k,0)$ design for any $t$ and $k$ with $1\leq t\leq\nu$ and $0\leq k\leq\nu$. A $t$-$(\nu,k,\lambda)$ design with $t>k$ must have $\lambda=0$ and must be the design $(\mathcal{P},\emptyset)$. These designs are called trivial designs. We have these conventions for the easiness of description in the sequel. A $t$-$(\nu,k,\lambda)$ design $(\mathcal{P},\mathcal{B})$ is also said to be trivial if every $k$-subset of $\mathcal{P}$ is a block.

A $t$-design is called simple if $\mathcal{B}$ does not contain repeated blocks. A $t$-$(\nu,k,\lambda)$ design is called a Steiner system and denoted by $S(t,k,\nu)$ if $t\geq 2$ and $\lambda=1$. The parameters of a $t$-$(\nu,k,\lambda)$ design satisfy:

[TABLE]

Let $\mathrm{GF}(q)$ denote the finite field with $q$ elements, where $q$ is a prime power. A linear code $\mathcal{C}$ over $\mathrm{GF}(q)$ may induce a $t$-design under certain conditions, which is formed by the supports of codewords of a fixed Hamming weight in $\mathcal{C}$. Let $\mathcal{P}(\mathcal{C})=\{0,1,\dots,\nu-1\}$ be the set of the coordinate positions of $\mathcal{C}$, where $\nu$ is the length of $\mathcal{C}$. For a codeword $\mathbf{c}=(c_{0},\dots,c_{\nu-1})$ in $\mathcal{C}$, the support of $\mathbf{c}$ is defined by

[TABLE]

Let $\mathcal{B}_{w}(\mathcal{C})=\frac{1}{q-1}\{\{\mathrm{Supp}(\mathbf{c}):wt(\mathbf{c})=w~{}\text{and}~{}\mathbf{c}\in\mathcal{C}\}\}$, here and hereafter $\{\{\}\}$ is the multiset notation and $\frac{1}{q-1}S$ denotes the multiset obtained after dividing the multiplicity of each element in the multiset $S$ by $q-1$. For some special $\mathcal{C}$, $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{w}(\mathcal{C})\right)$ is a $t$-design. If $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{w}(\mathcal{C})\right)$ is a $t$-design for any $0\leq w\leq\nu$, we say that the code $\mathcal{C}$ supports $t$-designs. Notice that such design $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{w}(\mathcal{C})\right)$ may have repeated blocks or may be simple or trivial.

With this approach, many $t$-designs have been obtained from linear codes [1, 13, 14, 17, 19, 20, 23, 26, 29, 30]. A major approach to constructing $t$-designs from codes is the use of the Assmus-Mattson Theorem [4, 22]. Another major approach to constructing $t$-designs from linear codes is the use of linear codes with $t$-homogeneous or $t$-transitive automorphism groups [15, Theorem 4.18]. Interplay between codes and designs could be found in [1, 2, 4, 12, 13, 14, 15, 16, 17, 22, 23, 24, 26, 29, 30].

In 2018, Ding, Munemasa and Tonchev [18] introduced a family of binary linear codes based on bent vectorial functions. These codes support $2$-designs, although they do not satisfy the conditions of the Assmus-Mattson theorem, and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general. Recently, Tang, Ding and Xiong [28] proved that some ternary codes, which do not satisfy the conditions of the Assmus-Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general, hold 2-designs. These works motivate us to develop a theory that can in one strike explain why these codes support $t$-designs on one hand, and may give new $t$-designs on the other hand.

In this paper, we first determine the parameters of some shortened and punctured codes of some codes supporting $t$-designs and pay special attention to the codes from bent functions and bent vectorial functions. Next, we give a characterization of codes supporting $t$-designs via the weight distributions of their shortened and punctured codes. Further, we present a generalization of the Assmus-Mattson theorem, which provides a unified explanation of the codes supporting $2$-designs in [18] and [28]. Finally, we present a design-theoretical characterization of differentially two-valued functions. Based on the established results, we use special differentially two-valued functions to give new binary linear codes, which hold $2$-designs but do not satisfy the conditions of the Assmus-Mattson theorem and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general.

The rest of this paper is arranged as follows. Section 2 introduces definitions and results related to linear codes, $t$-designs and differentially $\delta$-uniform functions. Section 3 investigates shortened and punctured codes of some linear codes supporting $t$-designs. Section 4 gives a characterization of codes supporting $t$-designs by means of their shorted and punctured codes. Section 5 presents a generalization of the Assmus-Mattson theorem. Section 6 gives a design-theoretical characterization of differentially two-valued functions and presents new codes that do not satisfy the conditions of the Assmus-Mattson theorem and do not admit $2$-transitive or $2$-homogeneous automorphism groups in general, but nevertheless hold $2$-designs. Section 7 concludes this paper and makes concluding remarks.

2 Preliminaries

In this section, we briefly recall some results on the Pless power moments of linear codes, $t$-designs, differentially $\delta$-uniform functions, and shortened and punctured codes.

2.1 The Pless power moments and the Assmus-Mattson theorem

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over the finite field $\mathrm{GF}(q)$, where $q$ is a prime power. Denote by $(A_{0},A_{1},\dots,A_{\nu})$ and $(A_{0}^{\perp},A_{1}^{\perp},\dots,A_{\nu}^{\perp})$ the weight distributions of $\mathcal{C}$ and its dual $\mathcal{C}^{\perp}$, respectively. The Pless power moments [22] are given by

[TABLE]

where $0\leq t\leq\nu$ and $S(t,j)=\frac{1}{j!}\sum_{i=0}^{j}(-1)^{j-i}\binom{j}{i}i^{t}$. These power moments can be employed to prove the following theorem [22, Theorem 7.3.1].

Theorem 2.1**.**

*Let $S\subseteq\{1,2,\dots,\nu\}$ with $\#S=s$. Then the weight distributions of $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are uniquely determined by $A_{1}^{\perp},\dots,A_{s-1}^{\perp}$ and the $A_{i}$ with $i\not\in S$. These values can be found from the first $s$ equations in (1). *

The following is a general version of the Assmus-Mattson Theorem.

Theorem 2.2**.**

*Let $\mathcal{C}$ be a linear code over $\mathrm{GF}(q)$ with length $\nu$ and minimum weight $d$. Let $\mathcal{C}^{\perp}$ with minimum weight $d^{\perp}$ denote the dual code of $\mathcal{C}$. Let $t~{}(1\leq t<\min\{d,d^{\perp}\})$ be an integer such that there are at most $d^{\perp}-t$ weights of $\mathcal{C}$ in $\{1,2,\ldots,\nu-t\}$. Then $(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C}))$ and $(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp}))$ are $t$-designs for all $k\in\{0,1,\ldots,\nu\}$. *

Notice that some of the designs in Theorem 2.2 may have repeated blocks or may be trivial in the senses defined in Section 1. The following lemma provides a criterion for obtaining a simple block set $\mathcal{B}_{k}(\mathcal{C})$ [15, Lemma 4.1].

Lemma 2.3**.**

Let $\mathcal{C}$ be a linear code over $\mathrm{GF}(q)$ with length $\nu$ and minimum weight $d$. Let $w$ be the largest integer with $w\leq\nu$ satisfying

[TABLE]

*Then there are no repeated blocks in $\mathcal{B}_{k}(\mathcal{C})$ for any $d\leq k\leq w$. Such a block set is said to be simple. *

Combining Theorem 2.2 and Lemma 2.3, one obtains the following Assmus-Mattson Theorem for constructing simple $t$-designs [3].

Theorem 2.4**.**

Let $\mathcal{C}$ be a linear code over $\mathrm{GF}(q)$ with length $\nu$ and minimum weight $d$. Let $\mathcal{C}^{\perp}$ with minimum weight $d^{\perp}$ denote the dual code of $\mathcal{C}$. Let $t~{}(1\leq t<\min\{d,d^{\perp}\})$ be an integer such that there are at most $d^{\perp}-t$ weights of $\mathcal{C}$ in the range $\{1,2,\ldots,\nu-t\}$. Then the following holds:

• •

$(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C}))$* is a simple $t$-design provided that $A_{k}\neq 0$ and $d\leq k\leq w$, where $w$ is defined to be the largest integer satisfying $w\leq\nu$ and*

[TABLE]

• •

$(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp}))$* is a simple $t$-design provided that $A_{k}^{\perp}\neq 0$ and $d^{\perp}\leq k\leq w^{\perp}$, where $w^{\perp}$ is defined to be the largest integer satisfying $w^{\perp}\leq\nu$ and*

[TABLE]

2.2 Shortened codes and punctured codes

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$ and $T$ a set of $t$ coordinate positions in $\mathcal{C}$. We use $\mathcal{C}^{T}$ to denote the code obtained by puncturing $\mathcal{C}$ on $T$, which is called the punctured code of $\mathcal{C}$ on $T$. Let $\mathcal{C}(T)$ be the subcode of $\mathcal{C}$, which is the set of codewords which are $\mathbf{0}$ on $T$. We now puncture $\mathcal{C}(T)$ on $T$, and obtain a linear code $\mathcal{C}_{T}$, which is called the shortened code of $\mathcal{C}$ on $T$. We will need the following result on the punctured and shortened codes of $\mathcal{C}$ [22, Theorem 1.5.7].

Lemma 2.5**.**

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $T$ be any set of $t$ coordinate positions. Then

(1) $\left(\mathcal{C}_{T}\right)^{\perp}=\left(\mathcal{C}^{\perp}\right)^{T}$ and $\left(\mathcal{C}^{T}\right)^{\perp}=\left(\mathcal{C}^{\perp}\right)_{T}$.

*(2) If $t<\min\{d,d^{\perp}\}$, then the codes $\mathcal{C}_{T}$ and $\mathcal{C}^{T}$ have dimension $m-t$ and $m$, respectively. *

2.3 Combinatorial t-designs and their intersection numbers

Let $\mathbb{D}=(\mathcal{P},\mathcal{B})$ be a $t$-$(\nu,k,\lambda)$ design. Let $T_{0}$ and $T_{1}$ be two disjoint subsets of $\mathcal{P}$ with $\#T_{0}=t_{0}$ and $\#T_{1}=t_{1}$. Denote by $\lambda_{T_{1}}^{T_{0}}$ the number of blocks in $\mathcal{B}$ that contain $T_{1}$ and are disjoint with $T_{0}$. These numbers $\lambda_{T_{1}}^{T_{0}}$ are called intersection numbers. For convenience, $\lambda_{T_{1}}^{\emptyset}$ and $\lambda_{\emptyset}^{T_{0}}$ are also written as $\lambda_{T_{1}}$ and $\lambda^{T_{0}}$ respectively. The next theorem will be useful in the sequel [27, Theorem 9.7].

Theorem 2.6**.**

Let $(\mathcal{P},\mathcal{B})$ be a $t$-$(\nu,k,\lambda)$ design. Let $T_{0},T_{1}\subseteq\mathcal{P}$, where $T_{0}\cap T_{1}=\emptyset$, $\#T_{0}=t_{0}$, $\#T_{1}=t_{1}$, and $t_{0}+t_{1}\leq t$. Then the intersection numbers $\lambda^{T_{0}}_{T_{1}}$ are independent of the specific choice of the elements in $T_{0}$ and $T_{1}$, and depend only on $t_{0}$ and $t_{1}$. Specifically,

[TABLE]

*where $\lambda(t_{0},t_{1})=\frac{\binom{\nu-t_{0}-t_{1}}{k-t_{1}}}{\binom{\nu-t}{k-t}}\lambda$. *

2.4 Differentially $\delta$-uniform functions

Let $F$ be a vectorial Boolean function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2^{m})$. If we use the function $F$ in a S-box of a cryptosystem, the efficiency of differential cryptanalysis is measured by the maximum of the cardinality of the set of elements $x$ in $\mathrm{GF}(2^{n})$ such that

[TABLE]

where $a\in\mathrm{GF}(2^{n})^{*}$ and $b\in\mathrm{GF}(2^{m})$. The function $F$ is called a differentially $\delta$-uniform function if

[TABLE]

where $\delta(a,b)=\#\{x\in\mathrm{GF}(2^{n}):F(x+a)+F(x)=b\}$. The function $F$ is said to be differentially two-valued if $\#\{\delta(a,b):a\in\mathrm{GF}(2^{n})^{*},b\in\mathrm{GF}(2^{m})\}=2$. The following result can be found in [6].

Proposition 2.7**.**

*Let $F$ be a differentially $\delta$-uniform function from $\mathrm{GF}(2^{n})$ to itself. Assume that $F$ is differentially two-valued. Then $\delta=2^{s}$ for some $s$, where $1\leq s\leq n$. *

Due to Proposition 2.7, we say that $F$ is differentially two-valued with $\{0,2^{s}\}$ if

[TABLE]

Results about differentially two-valued functions could be found in [10, 11]. When $n=m$, differentially $2$-uniform functions are also called almost perfect nonlinear (APN) functions.

For any function $F$ from $\mathrm{GF}(2^{n})$ to itself, the Walsh transform of $F$ at $(\lambda,\mu)\in\mathrm{GF}(2^{n})^{*}\times\mathrm{GF}(2^{n})$ is defined as

[TABLE]

where $\mathrm{Tr}_{2^{n}/2}(\cdot)$ is the absolute trace function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$. $\mathcal{W}_{F}(\lambda,\mu)$ are also called the Walsh coefficients of $F$. The component functions of $F$ are the Boolean functions $\mathrm{Tr}(\lambda F(x))$, where $\lambda\in\mathrm{GF}(2^{n})$. A component function $\mathrm{Tr}(\lambda F(x))$ is said to be bent if $\mathcal{W}_{F}(\lambda,\mu)=\pm 2^{\frac{n}{2}}$, for all $\mu\in\mathrm{GF}(2^{n})$. In this case, $\mathrm{Tr}(\lambda F(x))$ is also called a bent component of $F$.

3 Shortened and punctured codes of linear codes supporting $t$-designs

Linear codes supporting $t$-designs usually have special properties [15]. The automorphism group of such code may be $t$-transitive or $t$-homogeneous. Such code may satisfy the conditions in the Assmus-Mattson Theorems. Such code could be distance-optimal or dimension-optimal. In general, linear codes that support a $t$-design should have a certain kind of regularity. Hence, one would expect that some punctured and shortened codes of such linear code would be also attractive in certain sense. By puncturing or shortening such code, one may obtain linear codes with different parameters and interesting properties. This is one of the motivations of studying the punctured and shortened codes of linear codes supporting $t$-designs. A more important motivation is for developing a characterisation of $t$-designs supported by linear codes in Section 4.

In this section, we will first develop some general theory for some shortened and punctured codes of linear codes supporting $t$-designs, and will then use the general theory to determine the parameters and weight distributions of some shortened and punctured codes of two families of binary linear codes supporting $2$-designs.

3.1 General results for shortened and punctured codes of linear codes supporting

$t$-designs

In this subsection, we establish general results about shortened and punctured codes of linear codes supporting $t$-designs.

Recall that the binomial coefficient $\binom{a}{b}$ equals [math] when $a<b$ or $b<0$. Let $\mathcal{W}_{i}(\mathcal{C})$ denote the set of codewords of weight $i$ in a code $\mathcal{C}$ and $A_{i}(\mathcal{C})$ be the number of elements of $\mathcal{W}_{i}(\mathcal{C})$. We first give some results on parameters and the weight distributions of shortened codes and punctured codes of linear codes supporting $t$-designs.

Lemma 3.1**.**

Let $\mathcal{C}$ be a linear code of length $\nu$ and minimum distance $d$ over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $t$ and $k$ be two positive integers with $0<t<\min\{d,d^{\perp}\}$ and $1\leq k\leq\nu-t$. Let $T$ be a set of $t$ coordinate positions in $\mathcal{C}$. Suppose that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{i}(\mathcal{C})\right)$ is a $t$-design for all $i$ with $k\leq i\leq k+t$. Then

[TABLE]

Proof 3.2**.**

Let $\pi^{T}$ be the map from $\mathcal{C}$ to $\mathcal{C}^{T}$ defined as

[TABLE]

By Lemma 2.5, $\pi^{T}$ is a one-to-one linear transformation. Then

[TABLE]

where $\mu_{T_{1}}(\mathcal{W}_{k+t_{1}}(\mathcal{C}))$ is equal to the number of codewords in $\mathcal{W}_{k+t_{1}}(\mathcal{C})$ that satisfy the conditions $c_{i}=0$ if $i\in T\setminus T_{1}$ and $c_{i}\neq 0$ if $i\in T_{1}$. Note that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k+t_{1}}(\mathcal{C})\right)$ is a $t$-$(\nu,k+t_{1},\lambda)$ design with $\frac{1}{q-1}A_{k+t_{1}}(\mathcal{C})$ blocks, where $\lambda=\frac{\binom{k+t_{1}}{t}}{\binom{\nu}{t}}\frac{1}{q-1}A_{k+t_{1}}(\mathcal{C})$. Let $\lambda_{T_{1}}^{T\setminus T_{1}}$ be the intersection number of the $t$-design $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k+t_{1}}(\mathcal{C})\right)$. By Theorem 2.6, one has

[TABLE]

It then follows that

[TABLE]

Theorem 3.3**.**

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $t$ be a positive integer with $0<t<\min\{d,d^{\perp}\}$. Let $T$ be a set of $t$ coordinate positions in $\mathcal{C}$. Suppose that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{i}(\mathcal{C})\right)$ is a $t$-design for any $i$ with $d\leq i\leq\nu-t$. Then the shortened code $\mathcal{C}_{T}$ is a linear code of length $\nu-t$ and dimension $m-t$. The weight distribution $\left(A_{k}(\mathcal{C}_{T})\right)_{k=0}^{\nu-t}$ of $\mathcal{C}_{T}$ is independent of the specific choice of the elements in $T$. Specifically,

[TABLE]

Proof 3.4**.**

Let $\mathcal{C}(T)=\{(c_{i})_{i\in\mathcal{C}}\in\mathcal{C}:c_{i}=0\text{ for any }i\in T\}$. Let $\pi_{T}$ be the map from $\mathcal{C}(T)$ to $\mathcal{C}_{T}$ defined as

[TABLE]

By the definition of $\mathcal{C}(T)$ and $\mathcal{C}_{T}$, the map $\pi_{T}$ is a one-to-one linear transformation. Then

[TABLE]

where $\mu^{T}(\mathcal{W}_{k}(\mathcal{C}))$ is equal to the number of codewords in $\mathcal{W}_{k}(\mathcal{C})$ that satisfy the conditions $c_{i}=0$ if $i\in T$. Note that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t$-$(\nu,k,\lambda)$ design with $\frac{1}{q-1}A_{k}(\mathcal{C})$ blocks, where $\lambda=\frac{\binom{k}{t}}{\binom{\nu}{t}}\frac{1}{q-1}A_{k}(\mathcal{C})$. Let $\lambda^{T}$ be the intersection number of the $t$-design $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$. By Theorem 2.6, one has

[TABLE]

*The desired conclusion then follows from $A_{k}(\mathcal{C}_{T})=\mu^{T}(\mathcal{W}_{k}(\mathcal{C}))$ and Lemma 2.5. *

Theorem 3.5**.**

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $t$ be a positive integer with $0<t<d^{\perp}$. Let $T$ be a set of $t$ coordinate positions in $\mathcal{C}$. Suppose that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{i}(\mathcal{C})\right)$ is a $t$-design for any $i$ with $d\leq i\leq\nu$. Then the punctured code $\mathcal{C}^{T}$ is a linear code of length $\nu-t$ and dimension $m$. The weight distribution $\left(A_{k}(\mathcal{C}^{T})\right)_{k=0}^{\nu-t}$ of $\mathcal{C}^{T}$ is independent of the specific choice of the elements in $T$. Specifically,

[TABLE]

Proof 3.6**.**

*The desired results follow from Lemmas 2.5 and 3.1. *

Theorems 3.3 and 3.5 settle the parameters and weight distribution of the shortened code $\mathcal{C}_{T}$ and punctured code $\mathcal{C}^{T}$ of a code $\mathcal{C}$ supporting $t$-designs, respectively. In general it could be very hard to determine the weight distribution of a shortened or punctured code of a linear code.

3.2 Punctured and shortened codes of a family of binary codes

In this subsection, we determine the parameters and weight distributions of some punctured and shortened codes of a family of binary linear codes constructed from bent Boolean functions. As will be demonstrated shortly, the shortened and punctured codes are quite interesting.

Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, and let $D_{f}=\{d_{0},d_{1},\ldots,d_{\nu_{f}-1}\}\subseteq\mathrm{GF}(2^{n})$ be the support of $f$. Define a binary code of length $\nu_{f}$ by

[TABLE]

The following theorem on parameters of $\mathcal{C}(D_{f})$ was proved in [15, Theorems 14.13 and 14.15].

Theorem 3.7**.**

Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, where $n\geq 6$ and is even. Then $\mathcal{C}(D_{f})$ is a $[\nu_{f},n+1,(\nu_{f}-2^{\frac{n-2}{2}})/2]$ three-weight binary code with the weight distribution in Table 1 and it holds $2$-designs. The dual code $\mathcal{C}(D_{f})^{\perp}$ has minimum distance $4$.

Taking $T=\{t_{1}\}$, we have the parameters and the weight distribution of the shortened code $\mathcal{C}(D_{f})_{\{t_{1}\}}$ of $\mathcal{C}(D_{f})$ in the following theorem.

Theorem 3.8**.**

Let $t_{1}$ be an integer with $0\leq t_{1}<\nu_{f}$. Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, where $n\geq 6$ and is even. Then, the shortened code $\mathcal{C}(D_{f})_{\{t_{1}\}}$ is a two-weight binary linear code of length $\nu_{f}-1$ and dimension $n$, and has the weight distribution in Table 2.

Proof 3.9**.**

By Theorem 3.3,

[TABLE]

*The desired results follow from Theorem 3.7. *

Taking $T=\{t_{1},t_{2}\}$, we have the parameters and the weight distribution of the shortened code $\mathcal{C}(D_{f})_{\{t_{1},t_{2}\}}$ of $\mathcal{C}(D_{f})$ in the following theorem.

Theorem 3.10**.**

Let $t_{1}$ and $t_{2}$ be integers with $0\leq t_{1}<t_{2}<\nu_{f}$. Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, where $n\geq 6$ and is even. Then, the shortened code $\mathcal{C}(D_{f})_{\{t_{1},t_{2}\}}$ is a two-weight binary linear code of length $\nu_{f}-2$ and dimension $n-1$, and has the weight distribution in Table 3.

Proof 3.11**.**

By Theorem 3.3,

[TABLE]

*The desired results follow from Theorem 3.7. *

Taking $T=\{t_{1}\}$, we have the parameters and the weight distribution of the punctured code $\mathcal{C}(D_{f})^{\{t_{1}\}}$ of $\mathcal{C}(D_{f})$ in the following theorem.

Theorem 3.12**.**

Let $t_{1}$ be an integer with $0\leq t_{1}<\nu_{f}$. Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, where $n\geq 6$ and is even. Then, the punctured code $\mathcal{C}(D_{f})^{\{t_{1}\}}$ is a five-weight binary linear code of length $\nu_{f}-1$ and dimension $n+1$, and has the weight distribution in Table 4.

Proof 3.13**.**

By Theorem 3.5, for $k=\frac{\nu_{f}}{2}\pm 2^{\frac{n-4}{2}}$, one has

[TABLE]

and

[TABLE]

*The desired results follow from Theorem 3.7. *

Taking $T=\{t_{1},t_{2}\}$, we have the parameters and the weight distribution of the punctured code $\mathcal{C}(D_{f})^{\{t_{1},t_{2}\}}$ of $\mathcal{C}(D_{f})$ in the following theorem.

Theorem 3.14**.**

Let $t_{1},t_{2}$ be integers with $0\leq t_{1}<t_{2}<\nu_{f}$. Let $f$ be a bent function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2)$, where $n\geq 6$ and is even. Then, the punctured code $\mathcal{C}(D_{f})^{\{t_{1},t_{2}\}}$ is a seven-weight binary linear code of length $\nu_{f}-2$ and dimension $n+1$, and has the weight distribution in Table 5.

Proof 3.15**.**

By Theorem 3.5, for $k=\frac{\nu_{f}}{2}\pm 2^{\frac{n-4}{2}}$, one has

[TABLE]

[TABLE]

and

[TABLE]

*The desired results follow from Theorem 3.7 and Theorem 3.10. *

Example 3.16**.**

Let $\mathrm{GF}(2^{6})=\mathrm{GF}(2)[u]/\left(u^{6}+u^{4}+u^{3}+u+1\right)$ and $\alpha\in\mathrm{GF}(2^{6})$ such that $\alpha^{6}+\alpha^{4}+\alpha^{3}+\alpha+1=0$. Then $\alpha$ is a primitive element of $\mathrm{GF}(2^{6})$ and $f(x)=\mathrm{Tr}_{2^{6}/2}(\alpha x^{3})$ is a bent function on $\mathrm{GF}(2^{6})$ with $\nu_{f}=\#D_{f}=36$. $\mathcal{C}(D_{f})$ is a $[36,7,16]$ linear code with weight enumerator $1+63z^{16}+63z^{20}+z^{36}$.

Let $t_{1}$ be an integer with $0\leq t_{1}\leq 35$. Then the shortened code $\mathcal{C}(D_{f})_{\{t_{1}\}}$ has parameters $[35,6,16]$ and weight enumerator $1+35z^{16}+28z^{20}$. The punctured code $\mathcal{C}(D_{f})^{\{t_{1}\}}$ has parameters $[35,7,15]$ and weight enumerator $1+28z^{15}+35z^{16}+35z^{19}+28z^{20}+z^{35}$. The code $\mathcal{C}(D_{f})_{\{t_{1}\}}$ is optimal and the code $\mathcal{C}(D_{f})^{\{t_{1}\}}$ is almost optimal with respect to the Griesmer bound.

*Let $t_{1}$ and $t_{2}$ be two integers with $0\leq t_{1}<t_{2}\leq 35$. Then the shortened code $\mathcal{C}(D_{f})_{\{t_{1},t_{2}\}}$ has parameters $[34,5,16]$ and weight enumerator $1+19z^{16}+12z^{20}$. The punctured code $\mathcal{C}(D_{f})^{\{t_{1},t_{2}\}}$ has parameters $[34,7,14]$ and weight enumerator $1+12z^{14}+32z^{15}+19z^{16}+19z^{18}+32z^{19}+12z^{20}+z^{34}$. The code $\mathcal{C}(D_{f})_{\{t_{1},t_{2}\}}$ is optimal and the code $\mathcal{C}(D_{f})^{\{t_{1},t_{2}\}}$ is almost optimal with respect to the Griesmer bound. *

3.3 Punctured and shortened codes of another family of binary codes

In this subsection, we settle the parameters and weight distributions of some punctured and shortened codes of another family of binary linear codes constructed from bent vectorial Boolean functions. It will be shown that the shortened and punctured codes are interesting.

Let $F(x)$ be a vectorial function from $\mathrm{GF}(2^{n})$ to $\mathrm{GF}(2^{\ell})$. Let $\mathcal{C}(F)$ be the binary code of length $2^{n}$ defined by

[TABLE]

where $c_{a,b,c}(x)=\mathrm{Tr}_{2^{\ell}/2}(aF(x))+\mathrm{Tr}_{2^{n}/2}(bx)+c$.

The following was proved in [18, Theorem 5].

Theorem 3.17**.**

Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$, where $m\geq 3$. Then $\mathcal{C}(F)$ is a $[2^{2m},2m+\ell+1,2^{2m-1}-2^{m-1}]$ four-weight binary code with the weight distribution in Table 6. The dual code $\mathcal{C}(F)^{\perp}$ has minimum distance $4$.

Taking $T=\{t_{1}\}$, we have the parameters and the weight distribution of the shortened code $\mathcal{C}(F)_{\{t_{1}\}}$ of $\mathcal{C}(F)$ in the following theorem.

Theorem 3.18**.**

Let $t_{1}$, $m$ be integers with $0\leq t_{1}<2^{2m}$ and $m\geq 3$. Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$. Then, the shortened code $\mathcal{C}(F)_{\{t_{1}\}}$ is a binary linear code of length $2^{2m}-1$ and dimension $2m+\ell$, and has the weight distribution in Table 7.

Proof 3.19**.**

By Theorem 3.3,

[TABLE]

*The desired results follow from Theorem 3.17. *

Taking $T=\{t_{1},t_{2}\}$, we have the parameters and the weight distribution of the shortened code $\mathcal{C}(F)_{\{t_{1},t_{2}\}}$ of $\mathcal{C}(F)$ in the following theorem.

Theorem 3.20**.**

Let $t_{1}$, $t_{2}$ and $m$ be integers with $0\leq t_{1}<t_{2}<2^{2m}$ and $m\geq 3$. Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$. Then, the shortened code $\mathcal{C}(F)_{\{t_{1},t_{2}\}}$ is a binary linear code of length $2^{2m}-2$ and dimension $2m+\ell-1$, and has the weight distribution in Table 8.

Proof 3.21**.**

By Theorem 3.3,

[TABLE]

*The desired results follow from Theorem 3.17. *

Taking $T=\{t_{1}\}$, we have the parameters and the weight distribution of the punctured code $\mathcal{C}(F)^{\{t_{1}\}}$ of $\mathcal{C}(F)$ in the following theorem.

Theorem 3.22**.**

Let $t_{1}$ and $m$ be integers with $0\leq t_{1}<2^{2m}$ and $m\geq 3$. Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$. Then, the punctured code $\mathcal{C}(F)^{\{t_{1}\}}$ is a binary linear code of length $2^{2m}-1$ and dimension $2m+\ell+1$, and has the weight distribution in Table 9.

Proof 3.23**.**

By Theorem 3.5, for $k\in\left\{2^{2m-1}-2^{m-1},2^{2m-1},2^{2m-1}+2^{m-1}\right\}$, one has

[TABLE]

and

[TABLE]

*The desired results follow from Theorem 3.17. *

Taking $T=\{t_{1},t_{2}\}$, we have the parameters and the weight distribution of the punctured code $\mathcal{C}(F)^{\{t_{1},t_{2}\}}$ of $\mathcal{C}(F)$ in the following theorem.

Theorem 3.24**.**

Let $t_{1}$, $t_{2}$ and $m$ be integers with $0\leq t_{1}<t_{2}<2^{2m}$ and $m\geq 3$. Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$. Then, the punctured code $\mathcal{C}(F)^{\{t_{1},t_{2}\}}$ is a binary linear code of length $2^{2m}-2$ and dimension $2m+\ell+1$, and has the weight distribution in Table 10.

Proof 3.25**.**

By Theorem 3.5, for $k\in\left\{2^{2m-1}-2^{m-1},2^{2m-1},2^{2m-1}+2^{m-1}\right\}$, one has

[TABLE]

[TABLE]

and

[TABLE]

*The desired results follow from Theorem 3.17 and Theorem 3.20. *

Example 3.26**.**

Let $\mathrm{GF}(2^{6})=\mathrm{GF}(2)[u]/\left(u^{6}+u^{4}+u^{3}+u+1\right)$ and $\alpha\in\mathrm{GF}(2^{6})$ such that $\alpha^{6}+\alpha^{4}+\alpha^{3}+\alpha+1=0$. Then $F(x)=\mathrm{Tr}_{2^{6}/2^{3}}(\alpha x^{3})$ is a bent vectorial function from $\mathrm{GF}(2^{6})$ to $\mathrm{GF}(2^{3})$. The code $\mathcal{C}(F)$ is a $[64,10,28]$ linear code with weight enumerator $1+448z^{28}+126z^{32}+448z^{36}+z^{64}$.

Let $t_{1}$ be an integer with $0\leq t_{1}\leq 63$. Then the shortened code $\mathcal{C}(F)_{\{t_{1}\}}$ has parameters $[63,9,28]$ and weight enumerator $1+252z^{28}+63z^{32}+196z^{36}$. The punctured code $\mathcal{C}(F)^{\{t_{1}\}}$ has parameters $[63,10,27]$ and weight enumerator $1+196z^{27}+252z^{28}+63z^{31}+63z^{32}+252z^{35}+196z^{36}+z^{63}$. The code $\mathcal{C}(F)_{\{t_{1}\}}$ is optimal with respect to a one-step Griesmer bound, and $\mathcal{C}(F)^{\{t_{1}\}}$ has the same parameters as the best binary linear code known in the database maintained by Markus Grassl.

Let $t_{1}$ and $t_{2}$ be two integers with $0\leq t_{1}<t_{2}\leq 63$. Then the shortened code $\mathcal{C}(F)_{\{t_{1},t_{2}\}}$ has parameters $[62,8,28]$ and weight enumerator $1+140z^{28}+31z^{32}+84z^{36}$. The punctured code $\mathcal{C}(F)^{\{t_{1},t_{2}\}}$ has parameters $[62,10,26]$ and weight enumerator $1+84z^{26}+224z^{27}+140z^{28}+31z^{30}+64z^{31}+31z^{32}+140z^{34}+224z^{35}+84z^{36}+z^{62}$. The code $\mathcal{C}(F)_{\{t_{1},t_{2}\}}$ is optimal with respect to a one-step Griesmer bound, and $\mathcal{C}(F)^{\{t_{1},t_{2}\}}$ has the same parameters as the best binary linear code known in the database maintained by Markus Grassl.

4 Characterizations of linear codes supporting $t$-designs via shortened and punctured codes

In this section, we shall give a characterization of codes supporting $t$-designs in terms of their shortened and punctured codes. Let $\mathcal{P}$ be a set of $\nu$ elements and $\mathcal{B}$ a multiset of $k$-subsets of $\mathcal{P}$, where $1\leq k\leq\nu$. Let $\overline{\mathcal{B}}=\{\{\mathcal{P}\setminus B:B\in\mathcal{B}\}\}$.

Lemma 4.1**.**

*Let $(\mathcal{P},\mathcal{B})$ be a $(\nu-k)$-$(\nu,k,\lambda)$ design and $t$ an integer with $1\leq\nu-k\leq t\leq k$. Then $(\mathcal{P},\mathcal{B})$ is also a $t$-$\left(\nu,k,\binom{\nu-t}{\nu-k}\lambda/\binom{\nu-t}{k-t}\right)$ design. *

Proof 4.2**.**

Let $T$ be any $t$-subset of $\mathcal{P}$. It is observed that

[TABLE]

Then

[TABLE]

where $\lambda_{T}$ and $\lambda^{T^{\prime}}$ are the intersection numbers of the design $(\mathcal{P},\mathcal{B})$. By Theorem 2.6, one gets

[TABLE]

*It completes the proof. *

In the case of simple designs, Lemma 4.1 was known in the literature. The conclusion of Lemma 4.1 implies that a $(\nu-k)$-$(\nu,k,\lambda)$ design must be a trivial design, as every $k$-subset of the point set is a block of the design.

Lemma 4.3**.**

*Let $\mathbb{D}=(\mathcal{P},\mathcal{B})$ be a $t$-$(\nu,k,\lambda)$ design with $t\leq k\leq\nu-t$. Then $\overline{\mathbb{D}}=(\mathcal{P},\overline{\mathcal{B}})$ is a $t$-$(\nu,\nu-k,\overline{\lambda})$ design, where $\overline{\lambda}=\frac{\binom{\nu-t}{k}}{\binom{\nu-t}{k-t}}\lambda$. *

Proof 4.4**.**

*The desired results follow from Theorem 2.6. *

Lemma 4.5**.**

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$. Let $k$ and $t$ be two positive integers with $t\leq k\leq\nu-t$. Suppose that $A_{k}(\mathcal{C}_{T})$ is independent of the specific choice of the elements in $T$, where $T$ is any set of $t$ coordinate positions in $\mathcal{C}$. Let $\overline{\mathcal{B}}_{k}(\mathcal{C})=\frac{1}{q-1}\{\{\mathcal{P}(\mathcal{C})\setminus\mathrm{Supp}(\mathbf{c}):\mathbf{c}\in\mathcal{C},\mathrm{wt}(\mathbf{c})=k\}\}$. Then $\left(\mathcal{P}(\mathcal{C}),\overline{\mathcal{B}_{k}}(\mathcal{C})\right)$ is a $t$-$(\nu,\nu-k,\overline{\lambda})$ design, where $\overline{\lambda}=A_{k}(\mathcal{C}_{T})/(q-1)$. Further, $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t$-$(\nu,k,\lambda)$ design, where

[TABLE]

Proof 4.6**.**

Let $T=\{i_{1},\ldots,i_{t}\}$ be a subset of $\mathcal{P}(\mathcal{C})$. Note that

[TABLE]

if and only if

[TABLE]

if and only if

[TABLE]

*By assumption, $T$ is included in $A_{k}(\mathcal{C}_{T})/(q-1)$ blocks of $\overline{\mathcal{B}_{k}}(\mathcal{C})$, which is independent of the choices of the elements in $T$. This completes the proof of the first conclusion. The conclusion of the second part then follows from Lemma 4.3. *

The following theorem gives a characterization of codes supporting $t$-designs via the weight distributions of their shortened and punctured codes.

Theorem 4.7**.**

Let $\mathcal{C}$ be a $[\nu,m,d]$ linear code over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $t$ be a positive integer with $0<t<\min\{d,d^{\perp}\}$. Then the following statements are equivalent.

(1) $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t$-design for any $0\leq k\leq\nu$.

(2) $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ is a $t$-design for any $0\leq k\leq\nu$.

(3) For any $1\leq t^{\prime}\leq t$, the weight distribution $\left(A_{k}(\mathcal{C}_{T})\right)_{k=0}^{\nu-t^{\prime}}$ of the shortened code $\mathcal{C}_{T}$ is independent of the specific choice of the elements in $T$, where $T$ is any set of $t^{\prime}$ coordinate positions in $\mathcal{C}$.

*(4) For any $1\leq t^{\prime}\leq t$, the weight distribution $\left(A_{k}(\mathcal{C}^{T})\right)_{k=0}^{\nu-t^{\prime}}$ of the punctured code $\mathcal{C}^{T}$ is independent of the specific choice of the elements in $T$, where $T$ is any set of $t^{\prime}$ coordinate positions in $\mathcal{C}$. *

Proof 4.8**.**

(3) $\Longrightarrow$ (1): Suppose that the weight distribution $\left(A_{k}(\mathcal{C}_{T})\right)_{k=0}^{\nu-t^{\prime}}$ of the shortened code $\mathcal{C}_{T}$ is independent of the specific choice of the elements in $T$, where $1\leq t^{\prime}\leq t$. By Lemmas 4.3 and 4.5, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t^{\prime}$-design for any $0\leq k\leq\nu-t^{\prime}$. In particular, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t$-design for any $0\leq k\leq\nu-t$ and $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $(\nu-k)$-design for any $\nu-t+1\leq k\leq\nu-1$. By Lemma 4.1, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is also a $t$-design for any $\nu-t+1\leq k\leq\nu-1$. Since $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\nu}(\mathcal{C})\right)$ is always a $t$-design, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ is a $t$-design for any $0\leq k\leq\nu$.

(1) $\Longrightarrow$ (4): Recall that if $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\nu}(\mathcal{C})\right)$ is a $t$-design, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\nu}(\mathcal{C})\right)$ is also a $t^{\prime}$-design for $1\leq t^{\prime}\leq t$. The desired results follow from Theorem 3.5.

(4) $\Longrightarrow$ (2): By the condition in (4), Lemma 2.5 and the Pless power moments in (1), the weight distribution $\left(A_{k}((\mathcal{C}^{\perp})_{T})\right)_{k=0}^{\nu-t^{\prime}}$ of the shortened code $(\mathcal{C}^{\perp})_{T}$ is independent of the specific choice of the elements in $T$. Since Statement (3) implies Statement (1), the desired conclusion then follows.

*(2) $\Longrightarrow$ (3): By the condition in Item (2) and Theorem 3.5, the weight distribution $\left(A_{k}((\mathcal{C}^{\perp})^{T})\right)_{k=0}^{\nu-t^{\prime}}$ of the punctured code $(\mathcal{C}^{\perp})^{T}$ is independent of the specific choice of the elements in $T$, where $T$ is any set of $t^{\prime}$ coordinate positions in $\mathcal{C}^{\perp}$. The desired conclusion follows from Lemma 2.5 and the Pless power moments in (1). *

Notice that some of the $t$-designs $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ mentioned in Theorem 4.7 are trivial and some may not be simple.

Theorem 4.7 gives necessary and sufficient conditions for a code to support $t$-designs with $0<t<\min\{d,d^{\perp}\}$. It demonstrates the importance of the weight distribution of linear codes in the theory of $t$-designs, and will be used to develop a generalisation of the original Assmus-Mattson Theorem in the next section.

The following well-known result is clearly a corollary of Theorem 4.7. This demonstrates another usefulness of Theorem 4.7.

Corollary 4.9**.**

[24*, p.165]** Let $\mathcal{C}$ be a $[\nu,m,d]$ binary linear code with $m>1$, such that for each $w>0$ the supports of the codewords of weight $w$ form a $t$-design, where $t<d$. Then the supports of the codewords of each nonzero weight in $\mathcal{C}^{\perp}$ also form a $t$-design. *

5 A generalization of the Assmus-Mattson theorem

There is a strengthening of the Assmus-Mattson Theorem for special binary codes [8]. The objective of this section is to present another generalisation of the Assmus-Mattson Theorem documented in Theorem 2.2 and demonstrate its advantages over the original version.

5.1 Our generalisation of the Assmus-Mattson theorem

To develop the generalization of the Assmus-Mattson theorem, we need to prove the following lammas first.

Lemma 5.1**.**

Let $\mathcal{C}$ be a linear code of length $\nu$ over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $t$, $k$ be integers with $0\leq k\leq\nu$ and $0<t<\min\{d,d^{\perp}\}$. Let $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ be a $t$-$(\nu,k,\lambda_{k})$ design for some integer $\lambda_{k}$. Let $T$ be a set of $t$ coordinate positions in $\mathcal{C}$. Then

[TABLE]

Proof 5.2**.**

Let $\lambda^{T}$ be the number of blocks in $\mathcal{B}_{k}(\mathcal{C})$ that are disjoint with $T$. Then, $A_{k}(\mathcal{C}_{T})=(q-1)\lambda^{T}$. Using Theorem 2.6, one gets

[TABLE]

*It completes the proof. *

Lemma 5.3**.**

Let $\mathcal{C}$ be a linear code of length $\nu$ over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $s$ and $t$ be two positive integers with $0<t<\min\{d,d^{\perp}\}$. Let $T$ be a set of $t$ coordinate positions in $\mathcal{C}$. Suppose that $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{i}(\mathcal{C}^{\perp})\right)$ are $t$-$(\nu,i,\lambda^{\perp}_{i})$ designs for all $i$ with $0\leq i\leq s+t-1$. Then

[TABLE]

*where $0\leq k\leq s-1$ and $\lambda_{k+i}^{\perp}(t-i,i)=\frac{\binom{\nu-t}{k}}{\binom{\nu-t}{k-t+i}}\lambda_{k+i}^{\perp}$. *

Proof 5.4**.**

The desired results follow from Lemma 3.1 and the fact that

[TABLE]

Lemma 5.5**.**

*Let $\mathcal{C}$ be a $[\nu,m,d]$ code over $\mathrm{GF}(q)$ and $d^{\perp}$ the minimum distance of $\mathcal{C}^{\perp}$. Let $i_{1},\dots,i_{s}$ be $s$ positive integers and $T$ a set of $t$ coordinate positions of $\mathcal{C}$, where $0\leq i_{1}<\cdots<i_{s}\leq\nu-t$ and $1\leq t<\min\{d,d^{\perp}\}$. Suppose that $A_{i}(\mathcal{C}_{T})~{}(i\not\in\{i_{1},\dots,i_{s}\})$ and $A_{1}(\left(\mathcal{C}^{\perp}\right)^{T})$, $\dots$, $A_{s-1}(\left(\mathcal{C}^{\perp}\right)^{T})$ are independent of the elements of $T$. Then, the weight distribution of $\mathcal{C}_{T}$ is independent of the elements of $T$ and can be determined from the first $s$ equations in (1). *

Proof 5.6**.**

*By Lemma 2.5, $\mathcal{C}_{T}$ has dimension $m-t$, and $\left(\mathcal{C}_{T}\right)^{\perp}=\left(\mathcal{C}^{\perp}\right)^{T}$. The desired conclusions of this lemma then follow from Theorem 2.1. *

One of the main contributions of this paper is the following theorem, which generalizes the Assmus-Mattson theorem.

Theorem 5.7**.**

Let $\mathcal{C}$ be a linear code over $\mathrm{GF}(q)$ with length $\nu$ and minimum weight $d$. Let $\mathcal{C}^{\perp}$ denote the dual code of $\mathcal{C}$ with minimum weight $d^{\perp}$. Let $s$ and $t$ be two positive integers with $t<\min\{d,d^{\perp}\}$. Let $S$ be a $s$-subset of $\{d,d+1,\ldots,\nu-t\}$. Suppose that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\ell}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{\ell^{\perp}}(\mathcal{C}^{\perp})\right)$ are $t$-designs for $\ell\in\{d,d+1,\ldots,\nu-t\}\setminus S$ and $0\leq\ell^{\perp}\leq s+t-1$. Then $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ are $t$-designs for any $t\leq k\leq\nu$, and in particular,

• •

$\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$* is a simple $t$-design for all $k$ with $d\leq k\leq w$, where $w$ is defined to be the largest integer satisfying $w\leq\nu$ and*

[TABLE]

• •

and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ is a simple $t$-design for all $k$ with $d\leq k\leq w^{\perp}$, where $w^{\perp}$ is defined to be the largest integer satisfying $w^{\perp}\leq\nu$ and

[TABLE]

Proof 5.8**.**

For any $1\leq t^{\prime}\leq t$, let $S_{t^{\prime}}=S\cup\{i:\nu-t+1\leq i\leq\nu-t^{\prime}\}$ and $s^{\prime}=\#S_{t^{\prime}}$. Then, $s^{\prime}=s+t-t^{\prime}$. Then, the pair $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\ell}(\mathcal{C})\right)$ is $t^{\prime}$-design for any $\ell\in\{0,1,\ldots,\nu-t^{\prime}\}\setminus S_{t^{\prime}}$. By Lemma 5.1, $A_{i}(\mathcal{C}_{T})~{}(i\in\{0,1,\ldots,\nu-t^{\prime}\}\setminus S_{t^{\prime}})$ are independent of the elements of $T$, where $T$ is any set of $t^{\prime}$ coordinate positions of $\mathcal{C}$.

By the assumption of this theorem, the pair $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{\ell^{\perp}}(\mathcal{C}^{\perp})\right)$ is $t^{\prime}$-design for $0\leq\ell^{\perp}\leq(s^{\prime}+t^{\prime}-1)=(s+t-1)$. By Lemma 5.3, $A_{1}\left((\mathcal{C}^{\perp})^{T}\right)$, $\dots$, $A_{s^{\prime}-1}\left((\mathcal{C}^{\perp})^{T}\right)$ are independent of the elements of $T$, where $T$ is any set of $t^{\prime}$ coordinate positions of $\mathcal{C}$.

*By Lemma 5.5, the weight distribution of $\mathcal{C}_{T}$ is independent of the choice of the elements of $T$. It then follows from Theorem 4.7 that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ are $t$-designs for any $t\leq k\leq\nu$. The last conclusions on the simplicity of the designs $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ follow from Lemma 2.3. *

Notice that some of the $t$-designs from Theorem 5.7 are trivial, and some may not be simple. However, many of them are simple and nontrivial, and thus interesting.

We now show that Theorem 2.2 (i.e., the Assmus-Mattson Theorem) is a corollary of Theorem 5.7. To this end, we use Theorem 5.7 to derive Theorem 2.2.

Proof 5.9** (Proof of Theorem 2.2 using Theorem 5.7).**

*Let $w_{1},w_{2},\ldots,w_{s}$ be the nonzero weights of $\mathcal{C}$ in $\{d,d+1,\ldots,\nu-t\}$, where $s\leq d^{\perp}-t$. Put $S=\{w_{1},w_{2},\ldots,w_{s}\}$. Then $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\ell}(\mathcal{C})\right)$ is the trivial $t$-design $\left(\mathcal{P}(\mathcal{C}),\emptyset\right)$ for all $\ell\in\{d,d+1,\ldots,\nu-t\}\setminus S$. Note that $s+t-1\leq d^{\perp}-1$. Clearly, $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{\ell^{\perp}}(\mathcal{C}^{\perp})\right)$ are the trivial $t$-design $\left(\mathcal{P}(\mathcal{C}^{\perp}),\emptyset\right)$ for all $0\leq\ell^{\perp}\leq s+t-1$. It then follows from Theorem 5.7 that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ are $t$-designs for any $t\leq k\leq\nu$. Both $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{k}(\mathcal{C}^{\perp})\right)$ are clearly the trivial design $\left(\mathcal{P}(\mathcal{C}),\emptyset\right)$ for $0\leq k\leq t-1$, as we assumed that $t<\min\{d,d^{\perp}\}$. The desired conclusions of Theorem 2.2 then follow. *

One would naturally ask if Theorem 5.7 is more powerful than Theorems 2.2 and 2.4. The answer is yes, and this will be justified in the next subsection.

5.2 The extended Assmus-Mattson theorem can outperform the origianl one

The objective of this section is to show that Theorem 5.7 is more powerful than Theorems 2.2 and 2.4, and is indeed useful. To this end, we consider the linear codes investigated in [18] and [28].

In order for Theorem 5.7 to outperform the original Assmus-Mattson Theorem, one has to choose two positive integers $s$ and $t$ with $t<\min\{d,d^{\perp}\}$ and an $s$-subset $S$ of $\{d,d+1,\ldots,\nu-t\}$, and then prove that $\left(\mathcal{P}(\mathcal{C}),\mathcal{B}_{\ell}(\mathcal{C})\right)$ and $\left(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{\ell^{\perp}}(\mathcal{C}^{\perp})\right)$ are $t$-designs for $\ell\in\{d,d+1,\ldots,\nu-t\}\setminus S$ and $0\leq\ell^{\perp}\leq s+t-1$ with some other approach. Hence, extra work is needed when applying Theorem 5.7. This intuitively explains why Theorem 5.7 can outperform the original Assmus-Mattson The following two examples will clarify this statement.

Example 5.10**.**

Let $F$ be a bent vectorial function from $\mathrm{GF}(2^{2m})$ to $\mathrm{GF}(2^{\ell})$, where $m\geq 3$. Let $\mathcal{C}(F)$ be the code given in (2). By the weight distribution of $\mathcal{C}(F)$ in Table 6, for $k\not\in\{2^{2m-1},2^{2m-1}\pm 2^{m-1}\}$, the pair $(\mathcal{P}(\mathcal{C}(F)),\mathcal{B}_{k}(\mathcal{C}(F)))$ is a trivial $2$-design. By the definition of $\mathcal{C}(F)$, one has $\mathcal{B}_{2^{2m-1}}(\mathcal{C}(F))=\mathcal{B}_{2^{m-1}}(\mathrm{RM}_{2}(1,2m))$, where $\mathrm{RM}_{2}(1,2m)$ is the first order Reed-Muller code given by

[TABLE]

*It is well known that $\mathcal{B}_{2^{m-1}}(\mathrm{RM}_{2}(1,2m))$ holds $2$-design. Let $S=\{2^{2m-1}+2^{m-1},2^{2m-1}-2^{m-1}\}$. Then, the pair $(\mathcal{P}(\mathcal{C}(F)),\mathcal{B}_{k}(\mathcal{C}(F)))$ is a $2$-design for any $k\in\{0,1,\ldots,2^{2m}-2\}\setminus S$. Since $d((\mathcal{C}(F))^{\perp})=4$, the pair $(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{k}(\mathcal{C}(F)^{\perp}))$ is a trivial $2$-design for $0\leq k\leq 3=\#S+2-1$. Hence, by Theorem 5.7, the codes $\mathcal{C}(F)$ and $\mathcal{C}(F)^{\perp}$ support $2$-designs [18, Theorem 11]. The weight distribution of the code $\mathcal{C}(F)$ and Lemma 2.3 tell us that the $2$-designs supported by $\mathcal{C}(F)$ are simple. *

Example 5.11**.**

Let $m$ be an odd positive integer. Let $\mathcal{C}$ be the linear code defined by

[TABLE]

*where $\mathrm{Tr}_{3^{m}/3}(\cdot)$ is the trace function from $\mathrm{GF}(3^{m})$ to $\mathrm{GF}(3)$ and $\alpha$ is a generator of $\mathrm{GF}(3^{m})^{*}$. Then the code $\mathcal{C}$ have parameters $[\frac{3^{m}-1}{2},2m,3^{m-1}-3^{\frac{m-1}{2}}]$. Let $S=\left\{3^{m-1},3^{m-1}\pm 3^{\frac{m-1}{2}}\right\}$. Then, $A_{k}(\mathcal{C})=0$ if $k\not\in S\cup\{0\}$. Thus, the pair $(\mathcal{P}(\mathcal{C}),\mathcal{B}_{k}(\mathcal{C}))$ is a trivial $2$-design for any $k\in\{0,1,\ldots,\frac{3^{m}-1}{2}-2\}\setminus S$. According to [28, Corollary 1], $(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{4}(\mathcal{C}^{\perp}))$ is a Steiner system $S(2,4,\frac{3^{m}-1}{2})$ and is simple. It was known that $d(\mathcal{C}^{\perp})=4$ [28]. Thus the pair $(\mathcal{P}(\mathcal{C}^{\perp}),\mathcal{B}_{4}(\mathcal{C}^{\perp}))$ is a $2$-design for $0\leq k\leq 4=\#S+2-1$. Hence, by Theorem 5.7, the codes $\mathcal{C}$ and $\mathcal{C}^{\perp}$ support $2$-designs [28, Theorems 11 and 12]. The weight distribution of the code $\mathcal{C}$ and Lemma 2.3 tell us that the $2$-designs supported by $\mathcal{C}(F)$ are simple. *

The weight distributions of the codes in Examples 5.10 and 5.11 and the minimum distances of their duals are known. They tell us that the original Assumus-Mattson Theorems (i.e, Theorems 2.2 and 2.4) cannot be applied to prove that the codes in Examples 5.10 and 5.11 support $2$-designs. It is also known that the automorphism groups of these codes are not 2-transitive in general [18, 28]. However, Theorem 5.7 can do it. Therefore, Theorem 5.7 is more powerful than Theorems 2.2 and 2.4. Another application of Theorem 5.7 will be given in the next section.

6 $2$-designs and differentially $\delta$-uniform functions

Recall the definition of differentially $\delta$-uniform functions over $\mathrm{GF}(2^{n})$ and the notation introduced in Section 2.4. In this section, we shall give a connection between differentially $\delta$-uniform functions and $2$-designs, and present some new $2$-designs from some special differentially two-valued functions.

Let $F$ be a differentially $\delta$-uniform function over $\mathrm{GF}(2^{n})$. Define the following linear code

[TABLE]

It follows from Delsarte’s theorem [24] that the dual code $\mathcal{C}(F)^{\perp}$ of $\mathcal{C}(F)$ can be given by

[TABLE]

where $\mathbf{u}_{x}=(F(x),x,1)$. For any $x_{1},x_{2}\in\mathrm{GF}(2^{n})$ with $x_{1}\neq x_{2}$, denote by $\lambda_{\{x_{1},x_{2}\}}$ the cardinality of the set

[TABLE]

Let $a=x_{1}+x_{2}$ and $b=F(x_{1})+F(x_{2})$. Denote $E_{\{x_{1},x_{2}\}}=\{x\in\mathrm{GF}(2^{n}):F(x+a)+F(x)=b\}$. Then, $\delta(a,b)=\#(E_{\{x_{1},x_{2}\}})$ and

[TABLE]

where $x_{i}^{\prime}\in\mathrm{GF}(2^{n})$. Moreover, it is easily observed that

[TABLE]

where $\mathbf{c}_{i}=(c_{x})_{x\in\mathrm{GF}(2^{n})}$ with

[TABLE]

Consequently, one has

[TABLE]

So, we have proved the following theorem, which establishes a link between some $2$-designs and differentially two-valued functions.

Theorem 6.1**.**

*Let $F(x)$ be a function over $\mathrm{GF}(2^{n})$. Then $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp})\right)$ is a $2$-design if and only if $F$ is differentially two-valued. Furthermore, if $F$ is differentially two-valued with $\{0,2^{s}\}$, then $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp})\right)$ is a $2$-$(2^{n},4,2^{s-1}-1)$ design. *

Corollary 6.2**.**

*Let $F(x)$ be a function over $\mathrm{GF}(2^{n})$. Then $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp})\right)$ is a Steiner system $S(2,4,2^{n})$ if and only if $F$ is differentially two-valued with $\{0,4\}$. *

Magma program shows that the Steiner system $S(2,4,2^{n})$ from the differentially two-valued $\{0,4\}$ function $F(x)=x^{2^{2i}-2^{i}+1}$ [6, 21] or $F(x)=\alpha x^{2^{i}+1}+\alpha^{2^{m}}x^{2^{2m}+2^{m+i}}$ [7] is equivalent to the incidence structure from points and lines of the affine geometry $\mathrm{AG}(2^{\frac{n}{2}},\mathrm{GF}(4))$. It is still open whether there is a differentially two-valued $\{0,4\}$ function $F(x)$ such that $(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp}))$ is not equivalent to the Steiner system from affine geometry.

With Theorem 6.1, we can directly use results of the differentially two-valued functions to study the incidence structure $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp})\right)$. By Lemma 1 in [11] and Theorem 6.1, one has the following.

Corollary 6.3**.**

Let $F(x)$ be a differentially $\delta$-uniform function over $\mathrm{GF}(2^{n})$. Then $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{4}(\mathcal{C}(F)^{\perp})\right)$ forms a $2$-design if and only if

[TABLE]

Theorem 6.4**.**

*Let $F(x)$ over $\mathrm{GF}(2^{n})$ be differentially two-valued with $\{0,2^{s}\}$. Suppose that $\{\mathcal{W}_{F}(\lambda,\mu):\lambda\in\mathrm{GF}(2^{n})^{*},\mu\in\mathrm{GF}(2^{n})\}=\{0,2^{\frac{n+s}{2}},-2^{\frac{n+s}{2}}\}$. Then, the code $\mathcal{C}(F)$ and its dual $\mathcal{C}(F)^{\perp}$ support $2$-designs. *

Proof 6.5**.**

*Let $S=\left\{2^{n-1},2^{n-1}\pm 2^{\frac{n+s-2}{2}}\right\}$. Since $\mathcal{W}_{F}(\lambda,\mu)\in\{0,2^{\frac{n+s}{2}},-2^{\frac{n+s}{2}}\}$, the incidence structure $\left(\mathcal{P}(\mathcal{C}(F)),\mathcal{B}_{k}(\mathcal{C}(F))\right)$ forms a trivial $2$-design for any $k\not\in S$. It follows from Theorem 6.1 and $d(\mathcal{C}(E)^{\perp})\geq 4$ [9, Theorem 9] that the incidence structure $\left(\mathcal{P}(\mathcal{C}(F)^{\perp}),\mathcal{B}_{k}(\mathcal{C}(F)^{\perp})\right)$ forms a $2$-design for $0\leq k\leq 4=\#S+2-1$. The desired conclusions then follow from Theorem 5.7. *

Corollary 6.6**.**

Let $q$ be a power of $2$ and $m$ be a positive integer. Let $F(x)$ be a quadratic permutation over $\mathrm{GF}(q^{m})$ of the form

[TABLE]

*Suppose that $F(x)$ is differentially $q$-uniform. Then, the code $\mathcal{C}(F)$ and its dual $\mathcal{C}(F)^{\perp}$ support $2$-designs. *

Proof 6.7**.**

*By Theorems 5 and 6 in [25], the function $F(x)$ is differentially two-valued with $\{0,q\}$ and has Walsh coefficients in $\{0,\pm q^{\frac{m+1}{2}}\}$. The desired conclusion then follows from Theorem 6.4. *

To determine the parameters of the $2$-designs from the code $\mathcal{C}(F)$ and its dual $\mathcal{C}(F)^{\perp}$, we need the following lemma.

Lemma 6.8**.**

Let $F(x)$ be a function over $\mathrm{GF}(2^{n})$ with Walsh coefficients in $\{0,\pm 2^{\frac{n+s}{2}}\}$, where $0\leq s\leq n-1$. Then the code $\mathcal{C}(F)$ has parameters $[2^{n},2n+1,2^{n-1}-2^{\frac{n+s-2}{2}}]$ and its dual code $\mathcal{C}(F)^{\perp}$ has minimum distance

[TABLE]

Furthermore, the weight distribution of $\mathcal{C}(F)$ is given by

[TABLE]

*and $A_{i}=0$ for all other $i$. The number $A_{4}^{\perp}$ of the codewords of weight $4$ in $\mathcal{C}(F)^{\perp}$ equals to $\frac{2^{n-2}(2^{n}-1)(2^{s-1}-1)}{3}$. *

Proof 6.9**.**

Let $\mathbf{c}(a,b,c)=\left(\mathrm{Tr}\left(aF(x)+bx\right)+c\right)_{x\in\mathrm{GF}(2^{n})}$, where $a,b\in\mathrm{GF}(2^{n})$ and $c\in\mathrm{GF}(2)$. Then

[TABLE]

Then, $\mathrm{wt}(\mathbf{c}(a,b,c))\in\{0,2^{n},2^{n-1},2^{n-1}\pm 2^{\frac{n+s-2}{2}}\}$, and $\mathrm{wt}(\mathbf{c}(a,b,c))=0$ if and only if $a=b=c=0$. Thus, the dimension of $\mathcal{C}(F)$ is equal to $2n+1$. By Theorem 9 in [9], the minimal distance $d^{\perp}=4\text{ or }6$. Let $i_{1}=2^{n-1}-2^{\frac{n+s-2}{2}}$, $i_{2}=2^{n-1}$, and $i_{3}=2^{n-1}+2^{\frac{n+s-2}{2}}$. Note that $A_{2^{n}}=1$. The first three Pless power moments in (1) give

[TABLE]

Solving this system of equations, one gets

[TABLE]

Using the fourth Pless power moment in (1), one has

[TABLE]

Since $d^{\perp}=4$ or $6$, one obtains

[TABLE]

*It completes the proof. *

Combining Theorem 6.4 and Lemma 6.8, we deduce the following.

Theorem 6.10**.**

Let $F(x)$ over $\mathrm{GF}(2^{n})$ be differentially two-valued with $\{0,2^{s}\}$ and have Walsh coefficients in $\{0,2^{\frac{n+s}{2}},-2^{\frac{n+s}{2}}\}$. Then, $\mathcal{C}(F)$ holds a $2$-$(2^{n},k,\lambda)$ design for the following pair $(k,\lambda)$:

• •

$(k,\lambda)=\left(2^{n-1}\pm 2^{\frac{n+s-2}{2}},\left(2^{n-s-1}\pm 2^{\frac{n-s-2}{2}}\right)\left(2^{n-1}\pm 2^{\frac{n+s-2}{2}}-1\right)\right)$, and

• •

$(k,\lambda)=\left(2^{n-1},(2^{n-1}-1)(2^{n}-2^{n-s}+1)\right)$.

To show the existence of the $2$-designs in Theorem 6.4, we describe some functions over $\mathrm{GF}(2^{n})$ which are differentially two-valued with $\{0,2^{s}\}$ and have Walsh coefficients in $\{0,\pm 2^{\frac{n+s}{2}}\}$.

The first family of differentially two-valued monomials with Kasami exponents: $F(x)=x^{2^{2i}-2^{i}+1}$, where $n$ and $i$ are positive integers, $n\neq 3i$, $s=\gcd(n,i)$, and $\frac{n}{s}$ is odd. Then $F(x)$ is over $\mathrm{GF}(2^{n})$ and differentially two-valued with $\{0,2^{s}\}$, and has Walsh coefficients in $\{0,\pm 2^{\frac{n+s}{2}}\}$ [6, 21]. 2. 2.

The second family of differentially two-valued functions discovered by Bracken, Tan, and Tan [7]: $F(x)=\alpha x^{2^{i}+1}+\alpha^{2^{m}}x^{2^{2m}+2^{m+i}}$, where $n=3m$, $m$ and $i$ are two positive integers, $3\nmid m$, $3|(m+i)$, $s=\gcd(m,i)$, $2\nmid\frac{m}{s}$, and $\alpha$ is a primitive element of $\mathrm{GF}(2^{n})$. Then $F(x)$ is over $\mathrm{GF}(2^{n})$ and differentially two-valued with $\{0,2^{s}\}$, and has Walsh coefficients in $\{0,\pm 2^{\frac{3m+s}{2}}\}$ [7].

When $s\geq 2$, the original Assmus-Mattson Theorem says that the codes $\mathcal{C}(F)$ and $\mathcal{C}(F)^{\perp}$ for $F(x)=x^{2^{2i}-2^{i}+1}$ and $F(x)=\alpha x^{2^{i}+1}+\alpha^{2^{m}}x^{2^{2m}+2^{m+i}}$ support only $1$-designs. Magma program shows that, in general, the codes $\mathcal{C}(F)$ and $\mathcal{C}(F)^{\perp}$ are not $2$-transitive or $2$-homogeneous. However, by our generalization of the Assmus-Mattson theorem, these codes support $2$-designs. This is the third example showing that Theorem 5.7 is more powerful than the original Assmus-Mattson Theorems (i.e., Theorems 2.2 and 2.4).

7 Summary and concluding remarks

The main contributions of this paper are the following:

• •

The first one is the general theory for punctured and shorted codes of linear codes supporting $t$-design documented in Section 3.1. The general theory led to several classes of binary codes with interesting parameters and known weight distributions, which were presented in Sections 3.2 and 3.3. Some of the codes are distance-optimal and some have the best known parameters. These codes can be used for secret sharing [5, 31]. The general theory also played an important role in later sections.

• •

The second is the characterization of $t$-designs supported by a linear code via the weight distributions of punctured and shortened codes of the code, which was documented in Theorem 4.7. This characterization shows the importance of the weight distribution of linear codes in constructing $t$-designs from linear codes.

• •

The third is the generalized Assmus-Mattson theorem described in Theorem 5.7, which outperformed the original Assmus-Mattson Theorem in the three cases treated in this paper.

• •

The fourth is the link between some $2$-designs and differentially $\delta$-uniform functions, which was presented in Section 6. With is link, some $2$-designs and some Steiner systems $S(2,4,2^{n})$ were constructed.

It would be interesting to use the generalized Assmus-Mattson theorem (i.e., Theorem 5.7) to obtain more $t$-designs that cannot be produced with the original Assmus-Mattson theorem. The three cases dealt with in this paper are the only known ones to the best knowledge of the authors.