On deep holes of generalized projective Reed-Solomon codes

TL;DR

This paper characterizes deep holes in generalized projective Reed-Solomon codes over finite fields, linking their existence to specific sum conditions on subsets of the evaluation set, and extends understanding of decoding limits.

Contribution

It provides a complete characterization of deep holes in GPRS codes using sum conditions, and introduces new criteria involving polynomial degrees and subset sums for identifying deep holes.

Findings

Deep holes correspond to nonzero subset sums of elements in the evaluation set.

Characterization of deep holes for polynomials of degree k.

Special conditions for deep holes when the characteristic divides k.

Abstract

Determining deep holes is an important topic in decoding Reed-Solomon codes. Let $l\ge 1$ be an integer and $a_1,\ldots,a_l$ be arbitrarily given $l$ distinct elements of the finite field ${\bf F}_q$ of $q$ elements with the odd prime number $p$ as its characteristic. Let $D={\bf F}_q\backslash\{a_1,\ldots,a_l\}$ and $k$ be an integer such that $2\le k\le q-l-1$. In this paper, we study the deep holes of generalized projective Reed-Solomon code ${\rm GPRS}_q(D, k)$ of length $q-l+1$ and dimension $k$ over ${\bf F}_q$. For any $f(x)\in {\bf F}_q[x]$, we let $f(D)=(f(y_1),\ldots,f(y_{q-l}))$ if $D=\{y_1, ..., y_{q-l}\}$ and $c_{k-1}(f(x))$ be the coefficient of $x^{k-1}$ of $f(x)$. By using D\"ur's theorem on the relation between the covering radius and minimum distance of ${\rm GPRS}_q(D, k)$, we show that if $u(x)\in {\bf F}_q[x]$ with $\deg (u(x))=k$, then the received codeword $(u(D), c_{k-1}(u(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if the sum $\sum\limits_{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$. We show also that if $j$ is an integer with $1\leq j\leq l$ and $u_j(x):= \lambda_j(x-a_j)^{q-2}+\nu_j x^{k-1}+f_{\leq k-2}^{(j)}(x)$ with $\lambda_j\in {\bf F}_q^*$, $\nu_j\in {\bf F}_q$ and $f_{\leq{k-2}}^{(j)}(x)\in{\bf F}_q[x]$ being a polynomial of degree at most $k-2$, then $(u_j(D), c_{k-1}(u_j(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if and only if the sum $\binom{q-2}{k-1}(-a_j)^{q-1-k}\prod\limits_{y\in I}(a_j-y)+e$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$, where $e$ is the identity of the group ${\bf F}_q^*$. This implies that $(u_j(D), c_{k-1}(u_j(x)))$ is a deep hole of ${\rm GPRS}_q(D, k)$ if $p|k$.

Equations

$${\mathrm{GRS}}_q(D,k):=\{(f(x_1),\dots ,f(x_n))\in {\mathbf{F}}_q^n\mid f(x)\in {\mathbf{F}}_q[x],\mathrm{deg}f(x)\le k-1\}.$$

$${\mathrm{GPRS}}_q(D,k):=\{(f(x_1),\dots ,f(x_n),c_{k-1}(f(x)))\in {\mathbf{F}}_q^{n+1}\mid f(x)\in {\mathbf{F}}_q[x],\mathrm{deg}f(x)\le k-1\},$$

$${\mathrm{PPRS}}_q({\mathbf{F}}_q^{\ast },k):=\{(f(1),\dots ,f({\alpha }^{q-2}),c_{k-1}(f(x)))\in {\mathbf{F}}_q^q\mid f(x)\in {\mathbf{F}}_q[x],\mathrm{deg}f(x)\le k-1\},$$

$$d(u,v):=\#\{1\le i\le n\mid u_i\ne v_i,u_i\in {\mathbf{F}}_q,v_i\in {\mathbf{F}}_q\}.$$

$$d(C):=\min \{d(x,y)\mid x\in C,y\in C,x\ne y\},$$

$$d(u,C):=\underset{v\in C}{\min }\{d(u,v)\}.$$

$$\rho (C):=\max \{d(u,C)\mid u\in {\mathbf{F}}_q^n\}.$$

$$u(x):=\sum _{i=1}^n{u}_i\prod _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^n\frac{x-x_j}{x_i-x_j}\in {\mathbf{F}}_q[x],$$

$$n-\mathrm{deg}u(x)\le d(u,{\mathrm{GRS}}_q(D,k))\le n-k=\rho ({\mathrm{GRS}}_q(D,k)).$$

$$D:={\mathbf{F}}_q\backslash \{a_1,\dots ,a_l\}.$$

$$D:=\{y_1,\dots ,y_{q-l}\},$$

$$f(D):=(f(y_1),\dots ,f(y_{q-l})),$$

$${\mathrm{GPRS}}_q(D,k):=\{(f(D),c_{k-1}(f(x)))\in {\mathbf{F}}_q^{q-l+1}\mid f(x)\in {\mathbf{F}}_q[x],\mathrm{deg}f(x)\le k-1\}.$$

$$\underset{v\in C}{\min }\{d(u,v)\}=\rho (C).$$

$$\underset{v\in C}{\min }\{d(u+u_0,v)\}=\rho (C).$$

$$\{d(u+u_0,v)|v\in C\}=\{d(u+u_0,v+u_0)|v\in C\},$$

$$\underset{v\in C}{\min }\{d(u+u_0,v)\}=\underset{v\in C}{\min }\{d(u+u_0,v+u_0)\}.$$

$$\underset{v\in C}{\min }\{d(u+u_0,v)\}=\underset{v\in C}{\min }\{d(u,v)\}.$$

$$P_{k-1}:=\{f(x)\mid f(x)\in {\mathbf{F}}_q[x],\mathrm{deg}f(x)\le k-1\}.$$

$$d(u,{\mathrm{GPRS}}_q(D,k))=d(v,{\mathrm{GPRS}}_q(D,k)).$$

$$d(u,v)=d(u+w,v+w)=d(\lambda u,\lambda v)$$

$$d(u,{\mathrm{GPRS}}_q(D,k))$$

$$=$$

$$\displaystyle\left(\begin{array}[]{cc}1(D)&c_{k-1}(1)\\ x(D)&c_{k-1}(x)\\ \vdots&\vdots\\ x^{k-2}(D)&c_{k-1}(x^{k-2})\\ x^{k-1}(D)&c_{k-1}(x^{k-1})\end{array}\right)=\left(\begin{array}[]{cccc}1&\ldots&1&0\\ y_{1}&\ldots&y_{q-l}&0\\ \vdots&\vdots&\vdots&\vdots\\ y_{1}^{k-2}&\ldots&y_{q-l}^{k-2}&0\\ y_{1}^{k-1}&\ldots&y_{q-l}^{k-1}&1\end{array}\right)$$

$$V(\gamma_{1},\ldots,\gamma_{n}):=\det\left(\begin{array}[]{ccc}1&\dots&1\\ \gamma_{1}&\dots&\gamma_{n}\\ \vdots&\vdots&\vdots\\ \gamma_{1}^{n-1}&\dots&\gamma_{n}^{n-1}\end{array}\right).$$

$$V({\gamma }_1,\dots ,{\gamma }_n)=\prod _{1\le i<j\le n}({\gamma }_j-{\gamma }_i).$$

Full text

S. Hong is the corresponding author and was supported partially by National Science Foundation of China Grant # 11371260.

1. Introduction and the statements of the main results

Let ${\bf F}_{q}$ be the finite field of $q$ elements with $p$ as its characteristic. Let $n$ and $k$ be positive integers such that $k<n$. Let $D=\{x_{1},\cdots,x_{n}\}$ be a subset of ${\bf F}_{q}$, which is called the evaluation set. The generalized Reed-Solomon code ${\rm GRS}_{q}(D,k)$ of length $n$ and dimension $k$ over ${\bf F}_{q}$ is defined by:

[TABLE]

Moreover, the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ of length $n+1$ and dimension $k$ over ${\bf F}_{q}$ is defined as follows:

[TABLE]

where $c_{k-1}(f(x))$ is the coefficient of $x^{k-1}$ of $f(x)$. If $D={\bf F}_{q}^{*}$ , then it is called primitive projective Reed-Solomon code, namely,

[TABLE]

where $\alpha$ is a primitive element of ${\bf F}_{q}$. If $D={\bf F}_{q}$, then it is called the extended projective Reed Solomon code. For $u=(u_{1},\ldots,u_{n})\in{\bf F}_{q}^{n}$, $v=(v_{1},\ldots,v_{n})\in{\bf F}_{q}^{n}$, the Hamming distance $d(u,v)$ is defined by

[TABLE]

For any $[n,k]_{q}$ linear code $C$, the minimum distance $d(C)$ is defined by

[TABLE]

where $d(\cdot,\cdot)$ denotes the Hamming distance of two words. A linear $[n,k,d]$ code is called maximum distance separable (MDS) code if $d=n-k+1$. The error distance to code $C$ of a received word $u\in{\bf F}_{q}^{n}$ is defined by

[TABLE]

Clearly, $d(u,C)=0$ if and only if $u\in C$. The covering radius to code $C$ of a received word $u\in{\bf F}_{q}^{n}$ is defined by

[TABLE]

The most important algorithmic problem in coding theory is the maximum likelihood decoding (MLD): Given a received word $u\in{\bf F}_{q}^{n}$, find a word $v\in C$ such that $d(u,v)=d(u,C)$, then we decode $u$ to $v$ [7]. Therefore, it is very crucial to decide $d(u,C)$ for the word $u$. Guruswami and Sudan [3] provided a polynomial time list decoding algorithm for the decoding of $u$ when $d(u,C)\leq{n-\sqrt{nk}}$. When the error distance increases, Guruswami and Vardy [4] showed that that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We also notice that Dür [2] studied the Cauchy codes. In particular, Dür [2] got the relation between the covering radius and minimum distance of ${\rm GPRS}_{q}(D,k)$. When decoding the generalized projective Reed-Solomon code $C$,for a received word $u=(u_{1},\ldots,u_{n},u_{n+1})\in{\bf F}_{q}^{n+1}$, we define the Lagrange interpolation polynomial $u(x)$ of $u$ by

[TABLE]

i.e., $u(x)$ is the unique polynomial of degree $\deg u(x)\leq n-1$ such that $u(x_{i})=u_{i}$ for $1\leq i\leq n$ and $c_{k-1}(u(x))=u_{n+1}$. It is clear that $u\in C$ if and only if $d(u,C)=0$ if and only if $\deg u(x)\leq{k-1}$. Equivalently, $u\not\in C$ if and only if $d(u,C)\geq 1$ if and only if $k\leq\deg u(x)\leq n-1$. Evidently, we have the following simple bounds of $d(u,{\rm GRS}_{q}(D,k))$ which are due to Li and Wan.

Theorem 1.1. [6] Let $u$ be a received word such that $u\not\in{\rm GRS}_{q}(D,k)$. Then

[TABLE]

Let $u\in{\bf F}_{q}^{n}$. If $d(u,{\rm GRS}_{q}(D,k))=\rho({\rm GRS}_{q}(D,k))$, then the received word $u$ is called a deep hole of ${\rm GRS}_{q}(D,k)$. In 2007, Cheng and Murray [1] conjectured that a word $u$ is a deep hole of ${\rm GRS}_{q}({\bf F}_{q},k)$ if and only if $u(x)=ax^{k}+f_{\leq{k-1}}(x)$, where $u(x)$ is the Lagrange interpolation polynomial of the received word $u$ and $a\in{\bf F}_{q}^{*}$, $f_{\leq{k-1}}(x)\in{\bf F}_{q}[x]$ a polynomial of degree at most $k-1$. In 2012, Wu and Hong [9] disproved this conjecture by giving a new class of deep holes for Reed-Solomon codes ${\rm GRS}_{q}({\bf F}_{q}^{*},k)$. In fact, if $q\geq 4$ and $2\leq k\leq q-2$, then they showed that the received word $u$ is a deep hole if its Lagrange interpolation polynomial is $ax^{q-2}+f_{\leq{k-1}}(x)$. In [5], Hong and Wu proved that the received word $u$ is a deep hole of the generalized Reed-Solomon codes ${\rm GRS}_{q}(D,k)$ if its Lagrange interpolation polynomial is $\lambda(x-a_{i})^{q-2}+f_{\leq{k-1}}(x)$, where $\lambda\in{\bf F}_{q}^{*},a_{i}\in{\bf F}_{q}\backslash D$ and $f_{\leq{k-1}}(x)\in{\bf F}_{q}[x]$ a polynomial of degree at most $k-1.$

Throughout this paper, we always let $l$ be a positive integer and $a_{1},\ldots,a_{l}$ be any fixed $l$ distinct elements of ${\bf F}_{q}$. Let

[TABLE]

We write

[TABLE]

and for any $f(x)\in{\bf F}_{q}[x]$, we let

[TABLE]

and use $c_{k-1}(f(x))$ to denote the coefficient of $x^{k-1}$ of $f(x)$. Then we can rewrite the generalized projective Reed-Solomon code ${\rm GPRS}(D,k)$ with evaluation set $D$ as

[TABLE]

Let $u\notin{\rm GPRS}_{q}(D,k)$. If $d(u,{\rm GPRS}_{q}(D,k))=\rho({\rm GPRS}_{q}(D,k))$, then $u$ is also called a deep hole of generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$. In 2016, Zhang and Wan [10] studied the deep holes of projective Reed-Solomon code ${\rm GPRS}({\bf F}_{q},k)$. In fact, under the assumption that the only deep holes of ${\rm GRS}_{q}({\bf F}_{q},k)$ are those received codewords whose Lagrange interpolation polynomials are of degree $k$, they proved the following results by solving a subset sum problem.

Theorem 1.2. [10] *Let $q$ be an odd prime power. Assume that $3\leq k+1\leq p$ or $3\leq q-p+1\leq k+1\leq{q-2}$. Then the received codeword $(f({\bf F}_{q}),c_{k-1}(f(x)))$ with $\deg f(x)=k$ is a deep hole of ${\rm GPRS}({\bf F}_{q},k)$.

Theorem 1.3. [10] *Let $\deg f(x)\geq k+1$ and $s:=\deg f(x)-k+1$. If there are positive constants $c_{1}$ and $c_{2}$ such that $s<c_{1}\sqrt{q},(\frac{s}{2}+2)\log_{2}(q)<k<c_{2}q$, then $(f({\bf F}_{q}),c_{k-1}(f(x)))$ is not a deep hole of ${\rm GPRS}({\bf F}_{q},k)$.

In this paper, our main goal is to investigate the deep holes of the generalized projective Reed-Solomon codes ${\rm GPRS}_{q}(D,k)$. Actually, we will present characterizations for the received codewords of degrees $k$ and $q-2$ to be deep holes of generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$. The main results of this paper can be stated as follows.

Theorem 1.4. *Let $q$ be a prime power and $k$ and $l$ be positive integers such that $q\geq 5$ and $2\leq k\leq\min(q-3,q-l-1)$. Let $u(x)\in{\bf F}_{q}[x]$ with $\deg(u(x))=k$. Then the received codeword $(u(D),c_{k-1}(u(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm{\rm GPRS}}_{q}(D,k)$ if and only if the sum $\sum\limits_{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$.

Theorem 1.5. Let $q$ be a prime power and $k$ and $l$ be positive integers such that $q\geq 4$ and $2\leq k\leq{q-l-1}$. Let $j$ be an integer with $1\leq j\leq l$ and let $u_{j}(x):=\lambda_{j}(x-a_{j})^{q-2}+\nu_{j}x^{k-1}+f_{\leq k-2}^{(j)}(x)$ with $\lambda_{j}\in{\bf F}_{q}^{*}$, $\nu_{j}\in{\bf F}_{q}$ and $f_{\leq{k-2}}^{(j)}(x)\in{\bf F}_{q}[x]$ being a polynomial of degree at most $k-2$. Then the received codeword $(u_{j}(D),c_{k-1}(u_{j}(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if the sum $\binom{q-2}{k-1}a_{j}^{q-1-k}\prod\limits_{y\in I}(y-a_{j})+e$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$, where $e$ is the identity of the multiplicative group ${\bf F}_{q}^{*}$.

*Further, if $k\equiv 0\pmod{p}$, then the received codeword $(u_{j}(D),c_{k-1}(u_{j}(x)))$ is a deep hole of ${\rm GPRS}_{q}(D,k)$.

From Theorems 1.4 and 1.5, we can derive the following results on the deep holes of the primitive projective Reed-Solomon codes. Note that the proof of Theorem 1.6 relies also on a result on the zero subsets sum of the group ${\bf F}_{q}^{*}$ (see Lemma 2.8 below).

Theorem 1.6. *Let $q$ be an odd prime power such that $q\geq 5$ and $2\leq k\leq{q-3}$. If $u(x)=\lambda x^{k}+\gamma x^{k-1}+f_{\leq{k-2}}(x)$ with $\lambda\in{\bf F}_{q}^{*},\gamma\in{\bf F}_{q}$ and $f_{\leq{k-2}}(x)\in{\bf F}_{q}[x]$ being a polynomial of degree at most $k-2$, then the received codeword $(u({\bf F}_{q}^{*}),\gamma)$ is not a deep hole of the primitive projective Reed-Solomon code ${\rm PPRS}_{q}({\bf F}_{q}^{*},k)$.

Theorem 1.7. *Let $q\geq 4$ and $2\leq k\leq{q-2}$. If $u(x)=\lambda x^{q-2}+\delta x^{k-1}+f_{\leq{k-2}}(x)$ with $\lambda\in{\bf F}_{q}^{*},\delta\in{\bf F}_{q}$ and $f_{\leq{k-2}}(x)\in{\bf F}_{q}[x]$ being a polynomial of degree at most $k-2$, then the received codeword $(u({\bf F}_{q}^{*}),\delta)$ is a deep hole of the primitive projective Reed-Solomon code ${\rm PPRS}_{q}({\bf F}_{q}^{*},k)$.

In the proofs of Theorems 1.4 and 1.5, the basic tools are the MDS code and Vandemonde determinant. But we would also like to point out that a key ingredient in the proofs is Dür’s theorem on the relation between the covering radius and minimum distance of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ (see Lemma 2.6 below). Another important ingredient is a new result on the zero-sum problem in the finite field that we will prove in the next section.

This paper is organized as follows. First of all, in Section 2, we recall and prove several preliminary lemmas that are needed in the proof of Theorems 1.4 and 1.5. Consequently, in Section 3, we use the lemmas presented in Section 2 to give the proofs of Theorems 1.4 and 1.6. Finally, by using the results given in Section 2, we supply in Section 4 the proofs of Theorems 1.5 and 1.7.

2. Preliminary lemmas

In this section, our main aim is to prove several lemmas that are needed in the proof of Theorems 1.4 and 1.5. We begin with the following result on MDS codes.

Lemma 2.1. *Let $C$ be a MDS code and $u_{0}\in C$ be a given codeword. Then the received codeword $u$ is a deep hole of $C$ if and only if the received codeword $u+u_{0}$ is a deep hole of $C$.

Proof. First of all, let $u$ be a received codeword. Then by the definition of deep hole, one knows that $u$ is a deep hole of $C$ if and only if $d(u,C)=\rho(C)$ with $\rho(C)$ being the covering radius of $C$, if and only if

[TABLE]

Likewise, one has that the received codeword $u+u_{0}$ is a deep hole of $C$ if and only if

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

But $d(u+u_{0},v+u_{0})=d(u,v)$ for any codeword $u_{0}$. Hence (2.3) tells us that

[TABLE]

Now from (2.1), (2.2) and (2.4), one can deduce that $u$ is a deep hole of $C$ if and only if $u+u_{0}$ is a deep hole of $C$ as one desires. So Lemma 2.1 is proved. $\hfill{\Box}$

Remark 2.1. We should point out that if the codeword $u_{0}$ is not in $C$, then Lemma 2.1 is not true.

In what follows, we let

[TABLE]

We have the following result.

Lemma 2.2. Let $\#(D)=q-l$ and let $u\in{\bf F}_{q}^{q-l+1}$ and $v\in{\bf F}_{q}^{q-l+1}$ be two codewords with $u(x)$ and $v(x)$ being the Lagrange polynomials of $u$ and $v$. If $u(x)=\lambda v(x)+f_{\leq{k-2}}(x)$, where $\lambda\in{\bf F}_{q}^{*}$ and $f_{\leq{k-2}}(x)\in{\bf F}_{q}[x]$ is a polynomial of degree at most $k-2$, then

[TABLE]

*Further, $u$ is a deep hole of ${\rm GPRS}_{q}(D,k)$ if and only if $v$ is a deep hole of ${\rm GPRS}_{q}(D,k).$

Proof. Since $u(x)=\lambda v(x)+f_{\leq k-2}(x)$, we have $u(D)=\lambda v(D)+f_{\leq k-2}(D)$ and $c_{k-1}(u(x))=\lambda c_{k-1}(v(x))$. By the definition of Hamming distance, we know that for any code $C$ over ${\bf F}_{q}$, if $u$ and $v$ are codewords of $C$, then

[TABLE]

hold for any codeword $w$ of $C$ and any $\lambda\in{\bf F}_{q}^{*}$. Then from the definition of error distance and noticing that $u=(u(D),c_{k-1}(u(x)))$, we can deduce immediately that

[TABLE]

as required. The proof of Lemma 2.2 is complete. $\hfill{\Box}$

For a linear $[n,k]$ code $C$ with $n$ and $k$ being the length and dimension of $C$, respectively, we define the generator matrix, denoted by $G$, to be the $k\times n$ matrix of the form $G:=(g_{1},\ldots,g_{k})^{T}$, where $\{g_{1},\ldots,g_{k}\}$ is a basis of $C$ as a vector space. Since $D=\{y_{1},\ldots,y_{q-l}\}$, the following $k\times(q-l+1)$ matrix

[TABLE]

forms a generator matrix of ${\rm GPRS}_{q}(D,k)$. For the purpose of this paper, we will choose the above matrix as the generator matrix of ${\rm GPRS}_{q}(D,k)$.

Lemma 2.3. [8] *Let $C$ be an $[n,k]$ linear code and $G$ be the generator matrix of $C$. Then $C$ is a MDS code if and only if any $k$ distinct columns of $G$ are linear independent over finite field ${\bf F}_{q}$.

Throughout this paper, for any nonempty set $\{\gamma_{1},\ldots,\gamma_{n}\}\subset{\bf F}_{q}$, we define the Vandermonde determinant, denoted by $V(\gamma_{1},\ldots,\gamma_{n})$, as follows:

[TABLE]

We have the following well-known result.

Lemma 2.4. [8] One has

[TABLE]

In the following, we show that the generalized projective Reed-Solomon code is a MDS code.

Lemma 2.5. *Let $D\subset{\bf F}_{q}$. Then ${\rm GPRS}_{q}(D,k)$ is a $[q-l+1,k]$ MDS code over finite field ${\bf F}_{q}$.

Proof. Let $G$ be the generator matrix of ${\rm GPRS}_{q}(D,k)$ given in (2.5). Write $G:=(G_{1},\ldots,G_{q-l+1})$. Let $i_{1},\ldots,i_{k}$ be arbitrary $k$ distinct integers such that $1\leq i_{1}<\cdots<i_{k}\leq q-l+1.$ We claim that $\det(G_{i_{1}},\ldots,G_{i_{k}})\neq 0$ which will be proved in what follows.

If $i_{k}\leq q-l$, then it follows that

[TABLE]

The claim is true in this case.

If $i_{k}=q-l+1$, then by expanding the determinant according to the last column, we arrive at

[TABLE]

The claim is proved in this case.

Now by the claim, we can derive that any $k$ columns of the generator matrix $G$ is linear independent. Then ${\rm GPRS}_{q}(D,k)$ is a MDS code by Lemma 2.3.

This concludes the proof of Lemma 2.5. $\hfill{\Box}$

The following result about the relation between the covering radius and minimum distance of ${\rm GPRS}_{q}(D,k)$ will play a key role in this paper which is due to Dür [2].

Lemma 2.6. [2] Let $D$ be a proper subset of ${\bf F}_{q}$. Then

[TABLE]

The following result is well known.

Lemma 2.7. [11] *Let $G$ be a generator matrix of a MDS code $C=[n,k]$ over the finite field ${\bf F}_{q}$. If the covering radius $\rho(C)=n-k$, then a received codeword $u\in{\bf F}_{q}^{n}$ is a deep hole of $C$ if and only if the $(k+1)\times n$ matrix \left(\begin{array}[]{c}{G}\\ {u}\end{array}\right) can be served as the generator matrix of another MDS code.

In what follows, we show a result on the zero-sum problem in the finite field of odd characteristic.

Lemma 2.8. *Let $q=p^{s}$ with $p$ being an odd prime number and $k$ be an integer with $2\leq k\leq{q-3}$. Then there exist a subset $I\subseteq{\bf F}_{q}^{*}$ with $\#(I)=k$ such that $\sum\limits_{z\in I}z=0$.

Proof. Since $p$ is an odd prime number, it follows that for any $z\in{\bf F}_{q}^{*}$, one has $-z\in{\bf F}_{q}^{*}$ and $z\neq-z$ since $2z\neq 0$. But $|{\bf F}_{q}^{*}\setminus\{z,-z\}|=q-3\geq 2$ since $q\geq k+3\geq 5$. Now one can pick $z^{\prime}\in{\bf F}_{q}^{*}\setminus\{z,-z\}$. Then $-z^{\prime}\in{\bf F}_{q}^{*}\setminus\{z,-z,z^{\prime}\}$ since $2z^{\prime}\neq 0$. Continuing in this way, we finally arrive at

[TABLE]

We consider the following cases.

Case 1. $2\mid k$. In this case, we let $I=\{z_{1},-z_{1},\cdots,z_{\frac{k}{2}},-z_{\frac{k}{2}}\}$. Then $I\subset{\bf F}_{q}^{*}$ and we have

[TABLE]

as desired. Lemma 2.8 holds if $2\mid k$.

Case 2. $2\nmid k$. Then $k\geq 3$ and so $q\geq 7$ since $2\leq k\leq{q-3}$. We claim that there are three distinct elements $z^{\prime},z^{\prime\prime},z^{\prime\prime\prime}\in{\bf F}_{q}^{*}$ such that $z^{\prime}+z^{\prime\prime}+z^{\prime\prime\prime}=0$, which will be proved by dividing into the following three subcases.

Case 2.1. $p=3$. We pick a $z^{\prime}\in{\bf F}_{q}^{*}$. Then $3z^{\prime}=0$, $-z^{\prime}\neq 0$ and $2z^{\prime}\neq 0$. The latter implies that $z^{\prime}\neq-z^{\prime}$. Since $p=3$ and $q\geq 7$, we deduce that $q\geq 3^{2}=9$. Thus $|{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime}\}|=q-3\geq 6$. So we can choose a $z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime}\}$. But $2z^{\prime\prime}\neq 0$. Hence $-z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},z^{\prime\prime}\}$. It implies that $z^{\prime}+z^{\prime\prime}\neq 0$, namely, $z^{\prime}+z^{\prime\prime}\in{\bf F}_{q}^{*}$. Furthermore, we have that $z^{\prime}+z^{\prime\prime}$ is not equal to anyone of the four elements $z^{\prime},-z^{\prime},z^{\prime\prime}$ and $-z^{\prime\prime}$. That is, $z^{\prime}+z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},z^{\prime\prime},-z^{\prime\prime}\}$. Hence $-(z^{\prime}+z^{\prime\prime})\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},z^{\prime\prime},-z^{\prime\prime},z^{\prime}+z^{\prime\prime}\}$. Therefore there are three distinct elements $z^{\prime},z^{\prime\prime}$ and $-(z^{\prime}+z^{\prime\prime})$ in ${\bf F}_{q}^{*}$ such that their sum equals zero. The claim holds in this case.

Case 2.2. $p=5$. Take a $z^{\prime}\in{\bf F}_{q}^{*}$. Then $5z^{\prime}=0$ and none of $z^{\prime},2z^{\prime},3z^{\prime}$ and $4z^{\prime}$ equals zero. It follows that the four elements $z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime}$ are pairwise distinct. Since $q\geq 7>5$, one must have $q\geq 5^{2}=25$. Thus $|{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime}\}|=q-5\geq 20$. So we can choose $z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime}\}$. Then $-z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime}\}$ and $z^{\prime}+z^{\prime\prime}\neq 0$. The latter one tells us that $-(z^{\prime}+z^{\prime\prime})\in{\bf F}_{q}^{*}$. Obviously, $-z^{\prime\prime}\neq z^{\prime\prime}$ since $2z^{\prime\prime}\neq 0$. Hence $-z^{\prime\prime}\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime},z^{\prime\prime}\}$.

Furthermore, we can deduce that $z^{\prime}+z^{\prime\prime}$ is not equal to any of $z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime},z^{\prime\prime}$ and $-z^{\prime\prime}$. This infers that $-(z^{\prime}+z^{\prime\prime})\in{\bf F}_{q}^{*}\setminus\{z^{\prime},-z^{\prime},2z^{\prime},-2z^{\prime},z^{\prime\prime},-z^{\prime\prime},z^{\prime}+z^{\prime\prime}\}$ since $2(z^{\prime}+z^{\prime\prime})\neq 0$. Therefore we can find three distinct elements $z^{\prime},z^{\prime\prime}$ and $-(z^{\prime}+z^{\prime\prime})$ in ${\bf F}_{q}^{*}$ such that their sum equals zero. The claim holds in this case. The claim is proved in this case.

Case 2.3. $p\geq 7$. Then $le\neq 0$ for any integer $l$ with $1\leq l\leq 6$, where $e$ stands for the identity of the group ${\bf F}_{q}^{*}$. Since $e\neq 0,4e\neq 0$ and $5e\neq 0$, we have $e\neq 2e,e\neq-3e$ and $2e\neq-3e$. So there are three different elements $e,2e,-3e$ in ${\bf F}_{q}^{*}$ such that their sum is equal to zero as one desires. The claim is true in this case.

Now by the claim, we know that there are three integers $i_{1},i_{2}$ and $i_{3}$ such that $1\leq i_{1}<i_{2}<i_{3}\leq\frac{q-1}{2}$ and $z_{i_{1}}+z_{i_{2}}+z_{i_{3}}=0$.

If $q=7$, then letting $I=\{z_{i_{1}},z_{i_{2}},z_{i_{3}}\}$ gives us the desired result.

If $q>7$, then ${\bf F}_{q}^{*}\setminus\{\pm z_{i_{1}},\pm z_{i_{2}},\pm z_{i_{3}}\}$ is nonempty. By (2.6), we obtain that

[TABLE]

Since $2\nmid k$, $k-3$ is even. Evidently, the sum of the first $k-3$ elements on the right hand side of (2.7) is equal to zero because $z_{i}+(-z_{i})=0$ for all integers $1\leq i\leq\frac{q-1}{2}$. Then the first $k-3$ elements on the right hand side of (2.7) together with the three elements $z_{i_{1}},z_{i_{2}},z_{i_{3}}$ gives us the desired result. Thus Lemma 2.8 is true if $2\nmid k$.

This completes the proof of Lemma 2.8. $\hfill{\Box}$

For any positive integer $x$, we define its $p$-adic valuation, denoted by $v_{p}(x)$, to be the largest exponent $r$ such that $p^{r}$ divides $x$. In the conclusion of this section, We provide the following characterization on the divisibility of certain binomial coefficients by the prime number $p$.

Lemma 2.9. Let $q$ be a power of the odd prime $p$ and let $t$ be an integer with $2\leq t\leq q-1$. Then v_{p}\big{(}\binom{q-2}{t-1}\big{)}=v_{p}(t). Consequently, the binomial coefficient $\binom{q-2}{t-1}$ is divisible by $p$ if and only if $t$ is a multiple of $p$.

Proof. Clearly, one has

[TABLE]

Therefore

[TABLE]

Since $q$ is a power of $p$, it follows that for any positive integer $i$ with $i<q$, one has $v_{p}(i)<v_{p}(q)$, and so

[TABLE]

Then from (2.8) and (2.9) one derives that

[TABLE]

as required. It then follows that $p\mid\binom{q-2}{t-1}$ if and only if $p\mid t$. So Lemma 2.9 is proved. $\hfill{\Box}$

3. Proofs of Theorems 1.4 and 1.6

In this section, we use the lemmas presented in the previous to give the proofs of Theorems 1.4 and 1.6. At first, we show Theorem 1.4.

Proof of Theorem 1.4. Since $\deg(u(x))=k$, one may let $u(x)=\lambda x^{k}+\nu x^{k-1}+f_{\leq{k-2}}(x)$ with $\lambda\in{\bf F}_{q}^{*},\nu\in{\bf F}_{q}$ and $f_{\leq{k-2}}(x)\in{\bf F}_{q}[x]$ being a polynomial of degree at most $k-2$. Then $(u(D),c_{k-1}(u(x)))=(u(D),\nu)$. By Lemma 2.2, we have that $(u(D),\nu)$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if $(\lambda^{-1}u(D),\lambda^{-1}\nu)$ is a deep hole of ${\rm GPRS}_{q}(D,k)$. But

[TABLE]

where $w_{k}(x):=x^{k}$ and

[TABLE]

Then one has

[TABLE]

Since $\deg r_{k}(x)\leq k-1$, by the definition of ${\rm GPRS}_{q}(D,k)$ we have $(r_{k}(D),\lambda^{-1}\nu)\in{\rm GPRS}_{q}(D,k)$. Then it follows from Lemma 2.1 that $(\lambda^{-1}u(D),\lambda^{-1}\nu)$ is a deep hole of ${\rm GPRS}_{q}(D,k)$ if and only if $(w_{k}(D),0)$ is a deep hole of ${\rm GPRS}_{q}(D,k)$. Then we can deduce that $(u(D),c_{k-1}(u(x)))$ is a deep hole of ${\rm GPRS}_{q}(D,k)$ if and only if $(w_{k}(D),0)$ is a deep hole of ${\rm GPRS}_{q}(D,k)$.

We denote ${\bar{w}_{k}}:=(w_{k}(D),0)$. Let $G$ be the generator matrix of ${\rm GPRS}_{q}(D,k)$ as given in (2.5). Then we have

[TABLE]

Now we pick $k+1$ distinct integers with $1\leq j_{1}<\cdots<j_{k+1}\leq{q-l+1}$.

Case 1. $j_{k+1}\leq{q-l}$. Then one has

[TABLE]

Case 2. $j_{k+1}={q-l+1}$. We can compute and get that

[TABLE]

Now we introduce an auxiliary polynomial $g(y)$ as follows:

[TABLE]

Then Lemma 2.4 tells us that

[TABLE]

It infers that

[TABLE]

But

[TABLE]

Finally, (3.1) together with (3.2) and (3.3) gives us that

[TABLE]

By Lemma 2.5, we know that ${\rm GPRS}_{q}(D,k)$ is a $[q-l+1,k]$ MDS code which implies that

[TABLE]

Then by Lemma 2.6, one can deduce that

[TABLE]

It then follows immediately from Lemma 2.7 that $\bar{w}_{k}=(w_{k}(D),0)$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if the $(k+1)\times(q-l+1)$ matrix \left(\begin{array}[]{c}G\\ {\bar{w}_{k}}\end{array}\right) can be served as the generator matrix of a MDS code, if and only if any $k+1$ columns of \left(\begin{array}[]{c}G\\ {\bar{w}_{k}}\end{array}\right) are linear independent, if and only if for any $1\leq j_{1}<\cdots<j_{k+1}\leq q-l+1$, one has

[TABLE]

By the discussion in Cases 1 and 2, (3.4) tells us that (3.6) holds if and only if for any $1\leq j_{1}<\cdots<j_{k}\leq q-l$, one has $\sum\limits_{i=1}^{k}y_{j_{i}}\neq 0$. Hence we can derive that $(w_{k}(D),0)$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if the sum $\sum\limits_{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$ as desired.

Finally, we can conclude that $(u(D),c_{k-1}(u(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if the sum $\sum\limits_{y\in I}y$ is nonzero for any subset $I\subseteq D$ with $\#(I)=k$.

This finishes the proof of Theorem 1.4. $\hfill{\Box}$

We can now use Theorem 1.4 to show Theorem 1.6.

Proof of Theorem 1.6. Let $l=1$ and $a_{1}=0$. Then $D={\bf F}_{q}^{*}$. By Lemma 2.8, there exist a subset $I\subseteq{\bf F}_{q}^{*}$ with $\#(I)=k$ such that $\sum\limits_{y\in I}y=0$. It then follows from Theorem 1.4 that the received word $(u({\bf F}_{q}^{*}),c_{k-1}(u(x)))$=$(u({\bf F}_{q}^{*}),\gamma)$ is not a deep hole of the primitive projective Reed-Solomon code ${\rm PPRS}_{q}({\bf F}_{q}^{*},k)$. Therefore Theorem 1.6 is proved. $\hfill{\Box}$

4. Proofs of Theorems 1.5 and 1.7

In this section, we give the proof of Theorems 1.5 and 1.7. We begin with the proof of Theorem 1.5.

Proof of Theorem 1.5. First of all, we note that $j$ is an integer with $1\leq j\leq l$. We introduce a polynomial $f_{j}(x)$ as follows:

[TABLE]

and define a codeword ${\bar{f}_{j}}$ associated to $f_{j}(x)$ by

[TABLE]

Then $u_{j}(x)=\lambda_{j}f_{j}(x)+\nu_{j}x^{k-1}+f_{\leq k-2}^{(j)}(x)$ which implies that

[TABLE]

It follows from (4.1) that

[TABLE]

But

[TABLE]

and

[TABLE]

Hence

[TABLE]

It follows from Lemmas 2.1 and 2.2 that the received word $(u_{j}(D),c_{k-1}(u_{j}(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if $\bar{f}_{j}$ is a deep hole of ${\rm GPRS}_{q}(D,k)$.

Let $G$ be the generator matrix of ${\rm GPRS}_{q}(D,k)$ as given in (2.5). Since $y_{i}\neq a_{j}$ for all integers $i$ with $1\leq i\leq q-l$, we have $(y_{i}-a_{j})^{q-2}=(y_{i}-a_{j})^{-1}$. It then follows that

[TABLE]

On the other hand, from Lemma 2.7 we can deduce that $\bar{f}_{j}=(f_{j}(D),c_{k-1}(f_{j}(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$, if and only if \left(\begin{array}[]{c}G\\ {\bar{f}_{j}}\end{array}\right) generates a MDS code, by Lemma 2.3, if and only if any $k+1$ columns of \left(\begin{array}[]{c}G\\ {\bar{f}_{j}}\end{array}\right) are linear independent, if and only if for all $k+1$ integers $j_{1},\ldots,j_{k+1}$ with $1\leq j_{1}<\cdots<j_{k+1}\leq q-l+1$, one has

[TABLE]

In what follows, we choose arbitrarily $k+1$ integers $j_{1},\ldots,j_{k+1}$ such that $1\leq j_{1}<\cdots<j_{k+1}\leq q-l+1$. Consider the following two cases.

Case 1. $j_{k+1}\neq q-l+1$. Then $k+1\leq j_{k+1}\leq q-l$ and by (4.2), one has

[TABLE]

Thus one can deduce that

[TABLE]

[TABLE]

since $y_{j_{1}},\ldots,y_{j_{k+1}}$ are pairwise distinct.

Case 2. $j_{k+1}=q-l+1$. Then $1\leq j_{1}<\cdots<j_{k}\leq q-l$. From (4.2) and Lemma 2.4, we can deduce that

[TABLE]

Now from Cases 1 and 2, we can deduce by (4.4) that (4.3) holds for all $k+1$ integers $j_{1},\ldots,j_{k+1}$ with $1\leq j_{1}<\cdots<j_{k+1}\leq q-l+1$ if and only if for all integers $j_{1},\ldots,j_{k}$ with $1\leq j_{1}<\cdots<j_{k}\leq q-l$, one has

[TABLE]

which is equivalent to

[TABLE]

Since $f_{j}(x)=(x-a_{j})^{q-2}$, the binomial theorem gives us that

[TABLE]

Then one derives that (4.5) holds for all integers $j_{1},\ldots,j_{k}$ with $1\leq j_{1}<\cdots<j_{k}\leq q-l$ if and only if the following is true:

[TABLE]

or equivalently,

[TABLE]

since $q$ is odd. In other words, $\bar{f}_{j}=(f_{j}(D),c_{k-1}(f_{j}(x)))$ is a deep hole of the generalized projective Reed-Solomon code ${\rm GPRS}_{q}(D,k)$ if and only if the sum

[TABLE]

is nonzero for any subset $I\subseteq D$ with $\#(I)=k$. Hence the desired result follows immediately. The first part is proved.

Now we show the second part. Let $k\equiv 0\pmod{p}$. Then by Lemma 2.9, we have $\binom{q-2}{k-1}\equiv 0\pmod{p}$. So one can write $\binom{q-2}{k-1}=p\Delta$ with $\Delta$ being a positive integer. Then

[TABLE]

It then follows that

[TABLE]

for any subset $I\subseteq D$ with $\#(I)=k$. So it follows from the first part that the received codeword $(u_{j}(D),c_{k-1}(u_{j}(x)))$ is a deep hole of ${\rm GPRS}_{q}(D,k)$ if $k\equiv 0\pmod{p}$. The second part is proved.

The proof of Theorem 1.5 is complete. $\hfill{\Box}$

We can now present the proof of Theorem 1.7 as the conclusion of this paper.

Proof of Theorem 1.7. Letting $l=1$ and $a_{1}=0$ gives us that $D={\bf F}_{q}^{*}$.

If $p|k$, then by Theorem 1.5, one knows that the received word $(u({\bf F}_{q}^{*}),\delta)$ is a deep hole of the primitive projective Reed-Solomon code ${\rm PPRS}_{q}({\bf F}_{q}^{*},k)$.

If $p\nmid k$, then it follows from $a_{1}=0$ that

[TABLE]

for any subset $I\subseteq D$ with $\#(I)=k$. Hence $(u({\bf F}_{q}^{*}),\delta)$ is a deep hole of ${\rm PPRS}_{q}({\bf F}_{q}^{*},k)$.

This completes the proof of Theorem 1.7. $\hfill{\Box}$