Small weight code words arising from the incidence of points and hyperplanes in PG($\boldsymbol{n,q}$)

TL;DR

This paper improves bounds on small weight code words in the incidence code of points and hyperplanes in projective space, showing they are generated by hyperplanes through a fixed subspace for large q.

Contribution

It extends previous results by providing tighter bounds and characterizations of small weight code words in the projective space incidence code for larger q values.

Findings

Code words of weight ≤ (4q - √8q - 33/2)q^{n-2} are linear combinations of hyperplanes through a fixed (n-3)-space.

For q > 17 and q not in {25,27,29,31,32,49,121}, the structure of small weight code words is fully characterized.

The bounds on code word weights depend on the value of q, allowing for refined classifications.

Abstract

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.

Equations

$${\theta }_n=\frac{q^{n+1}-1}{q-1}\text{,}$$

$$H(P)=\{\begin{array}{l}1\text{if }P\in H\text{,}\\ 0\text{otherwise.}\end{array}$$

$$v\cdot w=\sum _{P\in \mathcal{P}(n,q)}v(P)w(P)\text{.}$$

$${C_{n-1}(n,q)}^{\bot}=\big{\{}v\in\mathcal{V}(n,q):\big{(}\forall c\in C_{n-1}(n,q)\big{)}\big{(}c\cdot v=0\big{)}\big{\}}\textnormal{.}$$

$$c(P)=\{\begin{array}{ll}a& \text{if}P=(0,1,a)\text{,}\\ b& \text{if}P=(1,0,b)\text{,}\\ -c& \text{if}P=(1,1,c)\text{,}\\ 0& \text{otherwise.}\end{array}$$

$$d=(\gamma c+\lambda m+{\lambda }^{\prime }{m}^{\prime }+{\lambda }^{\prime \prime }{m}^{\prime \prime }{)}^{\pi }$$

$$\text{wt}(d)=\{\begin{array}{l}3p-3\text{if}d(S)=0\text{,}\\ 3p-2\text{if}d(S)\ne 0\text{.}\end{array}$$

$$B_{n,q}=\begin{cases}\qquad\qquad 2q^{n-1}\quad&\textnormal{if }q<7\textnormal{ or }q\in\{8,9,16,25,27,49\}\textnormal{,}\\ \Big{(}3q-\sqrt{6q}-\frac{1}{2}\Big{)}q^{n-2}\quad&\textnormal{if }q\in\{7,11,13,17\}\textnormal{,}\\ \Big{(}3q-\sqrt{6q}+\frac{9}{2}\Big{)}q^{n-2}\quad&\textnormal{if }q\in\{19,121\}\textnormal{,}\\ \Big{(}4q-4\sqrt{q}-\frac{25}{2}\Big{)}q^{n-2}\quad&\textnormal{if }q\in\{29,31,32\}\textnormal{,}\\ \Big{(}4q-\sqrt{8q}-\frac{33}{2}\Big{)}q^{n-2}\quad&\textnormal{otherwise.}\\ \end{cases}$$

$$A_q=\{\begin{array}{ll}3q-3& \text{if }q\in \{7,11,13,17\}\text{,}\\ 3q+2& \text{if }q\in \{19,121\}\text{,}\\ 4q-21& \text{otherwise,}\end{array}\text{and}i=\{\begin{array}{ll}3& \text{if }q\in \{7,11,13,17\}\text{,}\\ 4& \text{otherwise.}\end{array}$$

$$\text{wt}(c)$$

$$\iff m$$

$$\Rightarrow q+2-i$$

$$\text{wt}(c_{|\pi })⩾j+(j-1)+\cdots +1=\frac{1}{2}j(j+1)\text{.}$$

$${\textnormal{wt}(c)}\geqslant\bigg{(}\frac{1}{2}j(j+1)-j\bigg{)}\theta_{n-2}+j\textnormal{.}$$

$$B_{n,q}\geqslant{\textnormal{wt}(c)}\geqslant\bigg{(}\frac{1}{2}q(q-1)-(q-1)\bigg{)}\theta_{n-2}+(q-1)\textnormal{,}$$

$$(3q-6){\theta }_{n-2}+2⩾\text{wt}(c)⩾(3q-3-3){\theta }_{n-2}+3\text{.}\text{qed}$$

$$|S|⩾(q^{n-1}-2q^{n-2})(q-3)+(2q^{n-1}-q^{n-2})+1=q^n-3q^{n-1}+5q^{n-2}+1.$$

$$|S^c|⩾(q^{n-1}-q^{n-2})(q-2)+1=q^n-3q^{n-1}+2q^{n-2}+1.$$

$${\theta }_n=|S|+|S^c|$$

$$=2q^n-6q^{n-1}+7q^{n-2}+2\text{,}$$

$$|S|{\theta }_{n-1}=|V|⩽2{\theta }_{n-1}+({\theta }_n-2)(2q^{n-2}+{\theta }_{n-3}).$$

$$\text{wt}(c)⩾{\theta }_{n-2}\cdot (q^2-q)+q=q^n\text{,}$$

$$\text{wt}(c)⩽{\theta }_{n-2}+(q+1)q^{n-2}={\theta }_{n-1}+q^{n-2}<2q^{n-1}\text{,}$$

$${\textnormal{wt}(c)}\geqslant q^{n-1}+(q-1)\cdot\big{(}q^{n-1}-\theta_{n-2}\big{)}=q^{n}-q^{n-1}+1\textnormal{,}$$

$$2q^{n-1}+{\theta }_{n-2}⩾\text{wt}(c)>q^{n-2}\cdot (2q-2)+({\theta }_{n-2}-q^{n-2})q^2+2\text{,}$$

$$(q^{n-2}-{\theta }_{n-3})(q-1)+1>\frac{1}{2}{\theta }_{n-1}$$

$$(3q-6){\theta }_{n-2}+3⩽\text{wt}(c)⩽B_{n,q}\text{.}$$

$$\text{wt}(c_{|\Sigma })$$

$$=\frac{1}{2}{q}^3-\frac{7}{2}q⩾\frac{1}{4}{q}^3\text{,}$$

$$\frac{1}{2}q(q-1)⩽\text{wt}(\sigma )$$

$$=q^2+q-y(q-3)\text{,}$$

$$\text{wt}(c_{|\sigma })$$

$$\displaystyle\geqslant\Big{(}\frac{1}{2}(q-5)\Big{)}\cdot\Big{(}\frac{1}{2}q(q-1)-3\Big{)}+\Big{(}\frac{1}{2}(q+6)\Big{)}\cdot(3q-3-3)+3$$

Full text

1 Preliminaries

Let $n\in\mathbb{N}$ and $q=p^{h}$, with $p$ prime and $h\in\mathbb{N}\setminus\{0\}$. Let PG($n,q$) be the $n$-dimensional Desarguesian projective space over the finite field of order $q$. In line with other articles, we define

[TABLE]

with the extension that $\theta_{m}=0$ if $m\in\mathbb{Z}\setminus\mathbb{N}$. Denote the set of all points of PG($n,q$) by $\mathcal{P}(n,q)$ and the set of all hyperplanes by $\mathcal{H}(n,q)$. Let $\mathcal{V}(n,q)$ be the $p$-ary vector space of functions from $\mathcal{P}(n,q)$ to $\mathbb{F}_{p}$; thus $\mathcal{V}(n,q)=\mathbb{F}_{p}^{\mathcal{P}(n,q)}$. Denote by $\boldsymbol{1}$ the function that maps all points to $1$.

Definition 1.0.1**.**

Let $v\in\mathcal{V}(n,q)$. Define the support of $v$ as ${\textnormal{supp}(v)}=\{P\in\mathcal{P}(n,q):v(P)\neq 0\}$ and the weight of $v$ as ${\textnormal{wt}(v)}=|{\textnormal{supp}(v)}|$. We will call all points of $\mathcal{P}(n,q)\setminus{\textnormal{supp}(v)}$ the holes of $v$.

We can identify each hyperplane $H\in\mathcal{H}(n,q)$ with the function $H\in\mathcal{V}(n,q)$ such that

[TABLE]

If a hyperplane $H$ is identified as a function, its representation as a vector will be called the incidence vector of the hyperplane $H$. It should be clear from the context whether we mean an actual hyperplane or such a function/vector.

Definition 1.0.2**.**

The $p$-ary linear code $C_{n-1}(n,q)$ is the subspace of $\mathcal{V}(n,q)$ generated by $\mathcal{H}(n,q)$, where we interpret the elements of the latter as functions in $\mathcal{V}(n,q)$. The elements of $C_{n-1}(n,q)$ are called code words.

Define the scalar product of two functions $v,w\in\mathcal{V}(n,q)$ as

[TABLE]

Definition 1.0.3**.**

We define the dual code of ${C_{n-1}(n,q)}$ as its orthogonal complement with respect to the above scalar product:

[TABLE]

Definition 1.0.4**.**

Let $v\in\mathcal{V}(n,q)$ and take a $k$-space $\kappa$ in PG($n,q$). If we let $\kappa$ play the role of PG($k,q$), we can naturally define the restriction of $v$ to the space $\kappa$ as the function ${v}_{|\kappa}\in\mathcal{V}(k,q)$ restricted to the point set $\mathcal{P}(k,q)\subseteq\mathcal{P}(n,q)$.

Definition 1.0.5**.**

Let $s$ be a line in PG($n,q$) and $v\in\mathcal{V}(n,q)$. If $s$ intersects ${\textnormal{supp}(v)}$ in $\alpha$ points ($0\leqslant\alpha\leqslant q+1$), we will call $s$ an $\alpha$-secant to ${\textnormal{supp}(v)}$. Furthermore,

• •

if $\alpha\leqslant 3$, $s$ will be called a short secant,

• •

if $\alpha\geqslant q-1$, $s$ will be called a long secant.

2 Known results

2.1 Results in general dimension

The minimum weight of the code $C_{n-1}(n,q)$ equals $\theta_{n-1}$. The code words corresponding to this weight are characterised.

Theorem 2.1.1** ([1, 10]).**

The code words of $C_{n-1}(n,q)$ having minimum weight are the scalar multiples of the incidence vectors of hyperplanes.

Bagchi and Inamdar [4, Theorem 1] gave a geometrical proof of this theorem, using blocking sets. Recently, Polverino and Zullo [12] characterised all code words up to the second smallest (non-zero) weight:

Theorem 2.1.2** ([12, 4]).**

Let $q=p^{h}$ with $p$ prime.

There are no code words of $C_{n-1}(n,q)$ with weight in the interval $]\theta_{n-1},2q^{n-1}[$. 2. 2.

The code words of weight $2q^{n-1}$ in $C_{n-1}(n,q)$ are the scalar multiples of the difference of the incidence vectors of two distinct hyperplanes of PG($n,q$).

So far, Theorem 2.1.2 summarises the best results known concerning the characterisation of small weight code words in $C_{n-1}(n,q)$ in case $n\geqslant 3$.

As a final note, we keep the following lemmata in mind.

Lemma 2.1.3** ([1, Chapter $6$], [12, Lemma $2$]).**

Let $c\in C_{n-1}(n,q)$, $c=\sum_{i}\alpha_{i}H_{i}$ for some $\alpha_{i}\in\mathbb{F}_{p}\setminus\{0\}$ and $H_{i}\in\mathcal{H}(n,q)$, and let $\kappa$ be a $k$-space of PG($n,q$), $1\leqslant\kappa\leqslant n$. Then $\kappa\cdot c=\sum_{i}\alpha_{i}$.

Lemma 2.1.4** ([12, Remark 3.1]).**

Let $c\in C_{n-1}(n,q)$ be a code word and $\kappa$ a $k$-space of PG($n,q$), $1\leqslant k\leqslant n$. Then ${c}_{|\kappa}$ is a code word of $C_{k-1}(k,q)$.

2.2 Results in the plane

Historically, most of the work done on this topic focuses on the planar case, i.e. the code of points and lines $C_{1}(2,q)$. Some early results on small weight code words in this particular code were those of Chouinard. In his PhD Thesis [5], he proved that, when $q=p$ prime, code words up to weight $2p$ are linear combinations of at most two lines. When $q=9$, he proved that code words having a weight in the interval $]q+1,2q[$ do not exist [6].

Fack et al. [8] improved the prime case. More specifically, these authors proved that, if $q=p\geqslant 11$, all code words of weight up to $2p+\frac{p-1}{2}$ are linear combinations of at most two lines. They cleverly made use of the existence of a Moorhouse base [11].

The prime case kept on inspiring more mathematicians. Next in line is Bagchi [3] on the one hand, and Szőnyi and Weiner [13] on the other hand (see Theorem 2.2.5). Bagchi proved the following:

Theorem 2.2.1** ([3, Theorem $1.1$]).**

Let $p\geqslant 5$. Then, the fourth smallest weight of $C_{1}(2,p)$ is $3p-3$. The only words of $C_{1}(2,p)$ of Hamming weight smaller than $3p-3$ are the linear combinations of at most two lines in the plane.

Bagchi knew this bound was sharp, as he discovered a code word of weight $3p-3$ which cannot be constructed as a linear combination of at most two lines when $p>3$ [2]. This code word was independently discovered by De Boeck and Vandendriessche [7] as well.

Example 2.2.2** ([2],[7, Example $10.3.4$]).**

Choose a coordinate system for PG($2,p$) and let $c$ be a vector of $\mathcal{V}(2,p)$, $p\neq 2$ a prime, such that

[TABLE]

Remark that ${\textnormal{supp}(c)}$ is covered by the three concurrent lines $m:X_{0}=0$, $m^{\prime}:X_{1}=0$ and $m^{\prime\prime}:X_{0}=X_{1}$.

The proof of $c$ being a code word of $C_{1}(2,p)$ relies on proving that $c$ belongs to ${C_{1}(2,p)}^{\bot}\subseteq C_{1}(2,p)$. As each of the three lines $m$, $m^{\prime}$ and $m^{\prime\prime}$ contains $p-1$ points with pairwise different, non-zero values, it is easy to see that such a code word can never be written as a linear combination of less than $p-1$ different lines.

As noted by Szőnyi and Weiner [13], the above example can be generalised as follows:

Example 2.2.3**.**

([13, Example $4.7$]) Let $c$ be the code word in Example 2.2.2, with corresponding lines $m$, $m^{\prime}$ and $m^{\prime\prime}$ considered as incidence vectors. Suppose $\pi$ is an arbitrary collineation of PG($2,p$) and let $\gamma\in\mathbb{F}_{p}\setminus\{0\}$ and $\lambda,\lambda^{\prime},\lambda^{\prime\prime}\in\mathbb{F}_{p}$. Then

[TABLE]

is a code word of weight $3p-3$ or $3p-2$, depending on the value of $\lambda+\lambda^{\prime}+\lambda^{\prime\prime}$.

By construction, it is easy to see that this generalised example has some interesting properties.

Proposition 2.2.4**.**

Suppose $d$ is the code word as constructed in Example 2.2.3. Let $S=(m\cap m^{\prime}\cap m^{\prime\prime})^{\pi}$. Then

[TABLE]

For somewhat larger values of $p$, Szőnyi and Weiner [13] improved Bagchi’s result:

Theorem 2.2.5** ([13, Theorem $4.8$ and Corollary $4.10$]).**

Let $c$ be a code word of $C_{1}(2,p)$, $p>17$ prime. If ${\textnormal{wt}(c)}\leqslant\max\{3p+1,4p-22\}$, then $c$ is either the linear combination of at most three lines or given by Example 2.2.3.

The same authors have proven the following results for $q$ not prime, proving for large values of $q$ that the code word described in Example 2.2.3 can only exist when $q$ is prime.

Theorem 2.2.6** ([13, Theorem $4.3$]).**

Let $c$ be a code word of $C_{1}(2,q)$, with $27<q$, $q=p^{h}$, $p$ prime. If

• •

${\textnormal{wt}(c)}<(\lfloor\sqrt{q}\rfloor+1)(q+1-\lfloor\sqrt{q}\rfloor)$, when $2<h$, or

• •

${\textnormal{wt}(c)}<\frac{(p-1)(p-4)(p^{2}+1)}{2p-1}$, when $h=2$,

then $c$ is a linear combination of exactly \big{\lceil}\frac{{\textnormal{wt}(c)}}{q+1}\big{\rceil} different lines.

We can now summarise these results concerning $C_{1}(2,q)$ in one corollary:

Corollary 2.2.7**.**

Let $c$ be a code word of $C_{1}(2,q)$, with $q=p^{h}$, $p$ prime, and $q\notin\{8,9,16,25,27,49\}$.

• •

If ${\textnormal{wt}(c)}\leqslant 3q-4$, then $c$ is a linear combination of at most two lines.

• •

If ${\textnormal{wt}(c)}\leqslant 3q+1$ and $q=121$, then $c$ is a linear combination of at most three lines.

• •

If ${\textnormal{wt}(c)}\leqslant\max\{3q+1,4q-22\}$ and $q>17$, $q\neq 121$, then $c$ is a linear combination of at most three lines or given by Example 2.2.3.

Proof.

If $q\leqslant 4$, then $3q-4\leqslant 2q$ and we can use Theorem 2.1.2. If $q>4$ and $q$ is prime, the proof immediately follows from Theorem 2.2.1 and Theorem 2.2.5.

Suppose $q>4$ is not prime. Then, by assumption, $q>27$, which means that $\max\{3q+1,4q-22\}=4q-22$. To apply Theorem 2.2.6, we only have to check the weight assumptions. One can verify that

• •

$4q-22<(\lfloor\sqrt{q}\rfloor+1)(q+1-\lfloor\sqrt{q}\rfloor)$ if $q\geqslant 10$,

• •

$3p^{2}+1<\frac{(p-1)(p-4)(p^{2}+1)}{2p-1}$ if $q=p^{2}\geqslant 121$,

• •

$4p^{2}-22<\frac{(p-1)(p-4)(p^{2}+1)}{2p-1}$ if $q=p^{2}\geqslant 144$.

We conclude that $c$ is a linear combination of at most $\lceil\frac{4q-22}{q+1}\rceil=4$ lines. If $c$ is a linear combination of precisely $4$ lines, then its weight is at least 4\cdot\big{(}(q+1)-3\big{)}=4q-8, a contradiction. ∎

3 The main theorem

Throughout this section, let $n\in\mathbb{N}\setminus\{0,1\}$ and $q=p^{h}$, with $p$ prime and $h\in\mathbb{N}\setminus\{0\}$. Let $c\in C_{n-1}(n,q)$ be an arbitrary code word. Furthermore, define

[TABLE]

This will be the assumed upper bound on the weight of $c$. By Theorem 2.1.2, we can always assume that $q\geqslant 7$ and $q\notin\{8,9,16,25,27,49\}$.

3.1 Preliminaries

Using Corollary 2.2.7 and Lemma 2.1.4, we can distinguish several types of small weight code words:

Definition 3.1.1**.**

Let $c$ be a code word, and $\pi$ a plane. We will call ${c}_{|\pi}$

• •

a code word of type $T_{w}$ if ${c}_{|\pi}$ is a linear combination of at most two lines, with $w$ the weight of ${c}_{|\pi}$.

• •

a code word of type ${T^{\textnormal{odd}}}$ if ${c}_{|\pi}$ is a code word as described in Example 2.2.3.

• •

a code word of type ${T^{\boldsymbol{\triangle}}}$ if ${c}_{|\pi}$ is a linear combination of three nonconcurrent lines.

• •

a code word of type ${T^{\bigstar}}$ if ${c}_{|\pi}$ is a linear combination of three concurrent lines.

• •

a code word of type $\boldsymbol{\mathcal{T}}=\{{T_{0}},{T_{q+1}},{T_{2q}},{T_{2q+1}},{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}},{T^{\bigstar}}\}$ if ${c}_{|\pi}$ is a code word of one of the types mentioned above.

• •

a code word of type $\boldsymbol{\mathcal{O}}$ if ${c}_{|\pi}$ is not a code word of one of the types mentioned above.

We will often make no distinction between the code word ${c}_{|\pi}$ and the plane $\pi$: if ${c}_{|\pi}$ is a code word of a certain type $T$, we will call $\pi$ a plane of type $T$.

Proposition 3.1.2**.**

If $\pi$ is a plane of type $\boldsymbol{\mathcal{T}}$ in PG($n,q$), then all lines of $\pi$ are either short or long secants to ${\textnormal{supp}(c)}$. If the type of $\pi$ is an element of $\{T_{0},T_{q+1},T_{2q},T_{2q+1}\}$, then all lines intersect ${\textnormal{supp}(c)}$ in at most $2$ or in at least $q$ points.∎

The following is a generalisation of Definition 3.1.1 to arbitrary dimension.

Definition 3.1.3**.**

Let $\Pi$ be a $k$-space of PG($n,q$), $2\leqslant k\leqslant n$. We will call the code word ${c}_{|\Pi}$ a code word of type $T\in\boldsymbol{\mathcal{T}}$ if the following is true:

• •

there exists a $(k-3)$-space $\kappa$ in $\Pi$ such that ${c}_{|\kappa}$ is a scalar multiple of $\boldsymbol{1}$.

• •

there exists a plane $\pi$ in $\Pi$ of type $T$, disjoint to $\kappa$.

• •

for all points $P\in\Pi\setminus\kappa$, c(P)=c\big{(}{\langle\kappa,P\rangle}\cap\pi\big{)}.

If ${c}_{|\Pi}$ is a code word of type $T$, we will often call the space $\Pi$ a space of type $T$ as well. If the type $T\in\boldsymbol{\mathcal{T}}$ is not known, we will call the code word ${c}_{|\Pi}$ (or the space $\Pi$) of type $\boldsymbol{\mathcal{T}}$. Remark that, if $k=2$, the above definition coincides with Definition 3.1.1.

The upcoming theorem is the main theorem of this article, which is an improvement of Theorem 2.1.2 when $q\geqslant 7$ and $q\notin\{8,9,16,25,27,49\}$:

Theorem 3.1.4**.**

Let $c$ be a code word of $C_{n-1}(n,q)$, with $n\geqslant 3$, $q$ a prime power and ${\textnormal{wt}(c)}\leqslant B_{n,q}$. Then $c$ can be written as a linear combination of incidence vectors of hyperplanes through a fixed $(n-3)$-space.

Corollary 3.1.5**.**

Let $c$ be a code word of $C_{n-1}(n,q)$, with $n\geqslant 3$, $q$ a prime power and ${\textnormal{wt}(c)}\leqslant B_{n,q}$. Then $c$ is a code word of type $\boldsymbol{\mathcal{T}}$.

Proof.

By Theorem 3.1.4, $c$ can be written as a linear combination of the incidence vectors of hyperplanes through a fixed $(n-3)$-space $\kappa$. If all planes disjoint to $\kappa$ are planes of type $\boldsymbol{\mathcal{O}}$, Lemma 3.2.1 implies that ${\textnormal{wt}(c)}\geqslant\frac{1}{2}q^{n-2}(q^{2}-3q-2)$, a contradiction for all $q$. All other properties of Definition 3.1.3 can easily be checked. ∎

3.2 The proof

3.2.1 The weight spectrum concerning lines and planes

In this subsection, we will prove some intermediate results, the first one stating that all lines contain few or many points of ${\textnormal{supp}(c)}$.

Lemma 3.2.1**.**

Suppose ${\textnormal{wt}(c)}\leqslant B_{n,q}$. Then

all lines are either short or long secants to ${\textnormal{supp}(c)}$. If $q\leqslant 17$, then all lines intersect ${\textnormal{supp}(c)}$ in at most $2$ or in at least $q$ points. 2. 2.

all planes of type $\boldsymbol{\mathcal{O}}$ contain at least $\frac{1}{2}q(q-1)$ points of ${\textnormal{supp}(c)}$.

Proof.

First of all, define the values

[TABLE]

Suppose, on the contrary, that $s$ is an $m$-secant to ${\textnormal{supp}(c)}$, with $i\leqslant m\leqslant q+2-i$. By Proposition 3.1.2 and Corollary 2.2.7, all planes through $s$ contain at least $A_{q}$ points of ${\textnormal{supp}(c)}$. We get

[TABLE]

From (1) and (2), we can conclude that all lines intersect ${\textnormal{supp}(c)}$ in at most $i-1$ or in at least $\frac{A_{q}\theta_{n-2}-{\textnormal{wt}(c)}}{\theta_{n-2}-1}$ points. Let $\pi\supseteq s$ be an arbitrary plane, thus ${\textnormal{wt}({c}_{|\pi})}\geqslant A_{q}$, and define $j=\min{\{{\textnormal{wt}({c}_{|l})}:l\subseteq\pi,{\textnormal{wt}({c}_{|l})}\geqslant i\}}$. Choose a point $P\in s\cap{\textnormal{supp}(c)}$. If all other $q$ lines in $\pi$ through $P$ contain at most $i-1$ points of ${\textnormal{supp}(c)}$, then ${\textnormal{wt}({c}_{|\pi})}\leqslant(i-2)q+m\leqslant(i-1)q-1<A_{q}$, a contradiction. Thus, through each point on $s\cap{\textnormal{supp}(c)}$ we find at least one line in $\pi$, other than $s$, containing at least $j$ points of ${\textnormal{supp}(c)}$. We find at least $m\geqslant j$ such lines, meaning that

[TABLE]

This holds for all planes through an $m$-secant with $i\leqslant m\leqslant q+2-i$, in particular for all planes through a $j$-secant in $\pi$. As such, we get

[TABLE]

When combining this result with $j\geqslant\frac{A_{q}\theta_{n-2}-{\textnormal{wt}(c)}}{\theta_{n-2}-1}$, we get a condition on ${\textnormal{wt}(c)}$, eventually leading to ${\textnormal{wt}(c)}>B_{n,q}$, a contradiction. We refer to Appendix A for the arithmetic details.

Let $\pi$ be a plane of type $\boldsymbol{\mathcal{O}}$. If no long secant is contained in this plane, ${\textnormal{wt}({c}_{|\pi})}\leqslant 2q+1<A_{q}$, a contradiction. Repeating the previous arguments, we get the same result as (3), for $j\geqslant q-1$. This concludes the proof. ∎

Using this result, we can deduce the following.

Lemma 3.2.2**.**

Suppose ${\textnormal{wt}(c)}\leqslant B_{n,q}$. If there exists a $(q-1)$-secant to ${\textnormal{supp}(c)}$, then there exists a $3$-secant to ${\textnormal{supp}(c)}$ as well.

Proof.

Let $s$ be a $(q-1)$-secant and suppose, on the contrary, that no $3$-secants exist. Remark that planes of type $\boldsymbol{\mathcal{T}}$ containing a $(q-1)$-secant always contain a $3$-secant. Hence, by Lemma 3.2.1, all planes through $s$ contain at least $\frac{1}{2}q(q-1)$ points of ${\textnormal{supp}(c)}$. We get

[TABLE]

which is a contradiction for all values of $q$. ∎

Lemma 3.2.3**.**

Suppose {\textnormal{wt}(c)}\leqslant\min\big{\{}(3q-6)\theta_{n-2}+2,B_{n,q}\big{\}}. Then all lines intersect ${\textnormal{supp}(c)}$ in at most $2$ or in at least $q$ points.

Proof.

By Lemma 3.2.1 and Lemma 3.2.2, it suffices to prove that $3$-secants to ${\textnormal{supp}(c)}$ cannot exist. Suppose there exists a $3$-secant to ${\textnormal{supp}(c)}$. By Corollary 2.2.7, all planes containing this $3$-secant have at least $3q-3$ points in common with ${\textnormal{supp}(c)}$. This gives us the following contradiction:

[TABLE]

3.2.2 Code words of weight $\boldsymbol{2q^{n-1}+\theta_{n-2}}$

In this section, we will prove Theorem 3.2.6, which essentially states that, if {\textnormal{wt}(c)}\leqslant\min\big{\{}(3q-6)\theta_{n-2}+2,B_{n,q}\big{\}}, the code word $c$ corresponds to a linear combination of at most two hyperplanes.

Lemma 3.2.4**.**

Assume that $S$ is a point set in PG($n,q$), $q\geqslant 7$, and every line intersects $S$ in at most $2$ or in at least $q$ points. Then one of the following holds:

1. (1)

$|S|\leqslant 2q^{n-1}+\theta_{n-2}$. 2. (2)

The complement of $S$, denoted by $S^{c}$, is contained in a hyperplane.

Proof.

We prove this by induction on $n$. Note that the statement is trivial for $n=1$, so assume that $n\geqslant 2$. Furthermore, we can inductively assume that for every hyperplane $\Pi$, either $|S\cap\Pi|\leqslant 2q^{n-2}+\theta_{n-3}$, in which case we call $\Pi$ a small hyperplane, or $S^{c}\cap\Pi$ is contained in an $(n-2)$-subspace of $\Pi$, in which case we call $\Pi$ a large hyperplane.

Case $1$: There exist two large hyperplanes $\Pi_{1}$ and $\Pi_{2}$, and a point $P\in S\setminus(\Pi_{1}\cup\Pi_{2})$.

Consider the lines through $P$. At most $q^{n-2}$ of these lines intersect $\Pi_{i}\setminus\Pi_{3-i}$ in a point of $S^{c}$, and $\theta_{n-2}$ of these lines intersect $\Pi_{1}\cap\Pi_{2}$. Hence, at least $\theta_{n-1}-2q^{n-2}-\theta_{n-2}=q^{n-1}-2q^{n-2}$ of these lines intersect $\Pi_{1}$ and $\Pi_{2}$ in distinct points of $S$. As $P\in S$, each of these lines contains at least three points of $S$. Therefore, they must contain at least $q$ points of $S$, thus at least $q-3$ points of S\setminus\big{(}\Pi_{1}\cup\Pi_{2}\cup\{P\}\big{)}. Since $\Pi_{1}\cup\Pi_{2}$ contains at least $2q^{n-1}-q^{n-2}$ points of $S$, we know that

[TABLE]

Case $2$: There exists a small hyperplane $\Pi$ and a point $P\in S^{c}\setminus\Pi$.

The small hyperplane $\Pi$ must contain at least $\theta_{n-1}-(2q^{n-2}+\theta_{n-3})=q^{n-1}-q^{n-2}$ points of $S^{c}$. Every line through $P$ and a point of $\Pi\cap S^{c}$ intersects $S^{c}$ in at least 2, thus in at least $q-1$ points of $S^{c}$. This yields that

[TABLE]

If both cases would occur simultaneously, then

[TABLE]

which is a contradiction if $q\geqslant 7$. Note that the existence of three large hyperplanes implies Case $1$, and the existence of two small hyperplanes implies Case $2$. Therefore, exactly one of these cases holds.

Assume that Case $1$ holds. Hence, Case $2$ cannot hold, so if there exists a small hyperplane, it has to contain the entirety of $S^{c}$ and the proof is done. As such, we can assume that all hyperplanes are large. Take a hyperplane $\Pi$. If the points of $S^{c}\cap\Pi$ span an $(n-2)$-space $\Sigma$, then $S^{c}\subseteq\Sigma\subseteq\Pi$. Otherwise, if a point $P\in S^{c}$ lies outside of $\Sigma$, ${\langle\Sigma,P\rangle}$ would be a (necessarily large) hyperplane, spanned by elements of $S^{c}$, a contradiction. In this way, we see that either some hyperplane contains all points of $S^{c}$, or for every hyperplane $\Pi$, $S^{c}\cap\Pi$ is contained in an $(n-3)$-subspace of $\Pi$. We can now use the same reasoning to prove that either some hyperplane contains all points of $S^{c}$, or for every hyperplane $\Pi$, $S^{c}\cap\Pi$ is contained in an $(n-4)$-space. Inductively repeating this process proves the theorem.

Assume that Case $2$ holds. Then there are at most two large hyperplanes, otherwise Case $1$ would hold. Consider the set $V=\{(P,\Pi):P\textnormal{ a point},\,\Pi\textnormal{ a hyperplane},\,P\in S\cap\Pi\}$. Counting the elements of $V$ in two ways yields

[TABLE]

Note that the right-hand side equals the exact size of $V$ in case $S$ is the union of two hyperplanes. Hence, the right-hand side equals $(2q^{n-1}+\theta_{n-2})\theta_{n-1}$. Thus, $|S|\leqslant 2q^{n-1}+\theta_{n-2}$. ∎

Lemma 3.2.5**.**

Suppose \theta_{n-1}<{\textnormal{wt}(c)}\leqslant\min\big{\{}(3q-6)\theta_{n-2}+2,B_{n,q}\big{\}}. Then there exists a $2$-secant to ${\textnormal{supp}(c)}$.

Proof.

Suppose that no $2$-secant to ${\textnormal{supp}(c)}$ exists and suppose $t$ is a $q$-secant to ${\textnormal{supp}(c)}$. By Corollary 2.2.7, all planes through $t$ containing at most $2q+1$ points of ${\textnormal{supp}(c)}$ correspond to planes of type ${T_{2q}}$. However, such planes contain several $2$-secants, contradicting the assumptions. Thus, by Lemma 3.2.3 and Lemma 3.2.4, all planes through $t$ must contain at least $q^{2}$ points of ${\textnormal{supp}(c)}$. In this way,

[TABLE]

which contradicts the weight assumptions. To conclude, all lines intersect ${\textnormal{supp}(c)}$ in [math], $1$ or $q+1$ points, which is only possible if ${\textnormal{supp}(c)}$ is a subspace. Once again, this contradicts the weight assumptions. ∎

Theorem 3.2.6**.**

Suppose {\textnormal{wt}(c)}\leqslant\min\big{\{}(3q-6)\theta_{n-2}+2,B_{n,q}\big{\}}. Then $c$ is a linear combination of the incidence vectors of at most two distinct hyperplanes.

Proof.

By Theorem 2.1.2, we may assume that 2q^{n-1}<{\textnormal{wt}(c)}\leqslant\min\big{\{}(3q-6)\theta_{n-2}+2,B_{n,q}\big{\}}. The proof will be done by induction on $n$. If $n=2$, Corollary 2.2.7 finishes the proof. Hence, let $n\geqslant 3$ and assume, for each hyperplane $\Pi$, that if {\textnormal{wt}({c}_{|\Pi})}\leqslant\min\big{\{}(3q-6)\theta_{n-3}+2,B_{n-1,q}\big{\}}, ${c}_{|\Pi}$ is a linear combination of at most two distinct $(n-2)$-subspaces of $\Pi$.

Suppose all hyperplanes contain at most $2q^{n-2}+\theta_{n-3}$ points of ${\textnormal{supp}(c)}$. Since ${\textnormal{supp}(c)}\neq\emptyset$, there must exist an $(n-2)$-space $\Pi_{n-2}$ intersecting ${\textnormal{supp}(c)}$ in $q^{n-2}$ or $\theta_{n-2}$ points, such that all hyperplanes through $\Pi_{n-2}$ contain either zero or $q^{n-2}$ points of ${\textnormal{supp}(c)}\setminus\Pi_{n-2}$. This yields

[TABLE]

a contradiction.

So consider a hyperplane $\Pi_{n-1}$, containing more than $2q^{n-2}+\theta_{n-3}$ points of ${\textnormal{supp}(c)}$. Due to Lemma 3.2.4, ${\textnormal{wt}({c}_{|\Pi_{n-1}})}\geqslant q^{n-1}$ and the holes of $\Pi_{n-1}$ are contained in an $(n-2)$-space $H_{n-2}$ of $\Pi_{n-1}$. By Lemma 3.2.5, we find a 2-secant $l$ to ${\textnormal{supp}(c)}$. Let $P$ and $Q$ be the points in $l\cap{\textnormal{supp}(c)}$ and let $\alpha=c(P)$.

Case $1$: $P,Q\notin\Pi_{n-1}$.

Suppose there is at most one hyperplane of type ${T_{2q}}$ or ${T_{2q+1}}$ through $l$. Fix an $(n-2)$-space $\Pi_{n-2}$ through $l$. By Lemma 3.2.4, at least $q$ hyperplanes through $\Pi_{n-2}$ each contain at least $q^{n-1}$ points of ${\textnormal{supp}(c)}$, thus

[TABLE]

which exceeds the imposed upper bound on ${\textnormal{wt}(c)}$ for all prime powers $q$, a contradiction.

Hence, we can choose a ${T_{2q}}$- or ${T_{2q+1}}$-typed hyperplane $\Sigma_{n-1}$ through $l$, different from the hyperplane ${\langle H_{n-2},l\rangle}$. Therefore, all holes in $\Sigma_{n-1}\cap\Pi_{n-1}$ are contained in the $(n-3)$-space $\Sigma_{n-1}\cap H_{n-2}$. As ${\textnormal{supp}({c}_{|\Sigma_{n-1}})}$ is the union or symmetric difference of precisely two $(n-2)$-subspaces and as $\Sigma_{n-1}\cap\Pi_{n-1}$ must be one of these two, the latter contains either $P$ or $Q$, contrary to the assumption of this case.

Case $2$: $P\in\Pi_{n-1}$.

Remark that, due to Lemma 3.2.4, ${\textnormal{wt}(c)}\leqslant 2q^{n-1}+\theta_{n-2}$. From this, we get that there are at least $q^{n-2}$ planes through $l$ containing at most $2q+1$ points of ${\textnormal{supp}(c)}$. Otherwise, we would have

[TABLE]

a contradiction whenever $q>2$.

The space $\Pi_{n-1}$ contains $\theta_{n-3}$ planes through a fixed line, so there exists a plane $\pi$ through $l$, not contained in $\Pi_{n-1}$, having at most $2q+1$ points of ${\textnormal{supp}(c)}$. If $Q\in\Pi_{n-1}$, we could choose another 2-secant lying in such an ‘external’ plane to $\Pi_{n-1}$ and replace $l$ (and $Q$ correspondingly) with this 2-secant. In this way, we may assume that $Q\in\pi\setminus\Pi_{n-1}$. Note that every line through $P$ containing at least two holes of $\Pi_{n-1}$ lies in $H_{n-2}$. Therefore, there are at most $\theta_{n-3}$ such lines through $P$. Every plane through $l$ intersects $\Pi_{n-1}$ in a line through $P$, hence there must be at least $q^{n-2}-\theta_{n-3}$ planes through $t$ of type ${T_{2q}}$ of ${T_{2q+1}}$, resulting in at least $q^{n-2}-\theta_{n-3}$ lines in $\Pi_{n-1}$, through $P$, each containing at least $q$ points all having the same non-zero value $\alpha$ in $c$. This yields at least

[TABLE]

points in $\Pi_{n-1}$ with value $\alpha$.

Now suppose, on the contrary, that $c$ is a code word of minimal weight such that $c$ cannot be written as a linear combination of at most two hyperplanes. Then ${\textnormal{wt}(c-\alpha\Pi_{n-1})}<{\textnormal{wt}(c)}$, thus the code word $c-\alpha\Pi_{n-1}$ is a linear combination of exactly two hyperplanes. As a consequence, $c$ must be a linear combination of precisely three hyperplanes, implying that ${\textnormal{wt}(c)}\geqslant 3(q^{n-1}-q^{n-2})$, contradicting the weight assumptions. ∎

3.2.3 Going higher on the weight spectrum

It will turn out that we can go further than the code words of weight $2q^{n-1}+\theta_{n-2}$. Moreover, we will be able to prove that a code word of weight at most $B_{n,q}$ corresponds to a linear combination of hyperplanes through a fixed $(n-3)$-space (Theorem 3.1.4).

Due to Theorem 3.2.6, we can assume the following on the weight of the code word $c$:

[TABLE]

As we are mainly interested in the case $n\geqslant 3$, the inequality above implies that $q\geqslant 29$, which we will keep in mind for the remainder of this section.

Lemma 3.2.7**.**

Suppose $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Then there exists a $3$-secant to ${\textnormal{supp}(c)}$.

Proof.

Suppose that there does not exist a $3$-secant to ${\textnormal{supp}(c)}$. By Lemma 3.2.1 and Lemma 3.2.2, all lines intersect ${\textnormal{supp}(c)}$ in at most $2$ or in at least $q$ points. Applying Lemma 3.2.4, we obtain that ${\textnormal{wt}(c)}\leqslant 2q^{n-1}+\theta_{n-2}$ or ${\textnormal{wt}(c)}\geqslant q^{n}$, contradicting our weight assumptions. ∎

Lemma 3.2.8**.**

Suppose $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Then all planes containing a $3$-secant are planes of type $\boldsymbol{\mathcal{T}}$.

Proof.

Suppose that $\sigma$ is a plane of type $\boldsymbol{\mathcal{O}}$ containing a $3$-secant $t$ and suppose that $\Sigma$ is a solid containing $\sigma$. We claim that ${\textnormal{wt}({c}_{|\Sigma})}\geqslant\frac{1}{4}q^{3}$.

In the first case, suppose that all planes in $\Sigma$ through $t$ are planes of type $\boldsymbol{\mathcal{O}}$. By Lemma 3.2.1,

[TABLE]

the last inequality being valid whenever $q>3$.

In the second case, suppose there exists a plane $\pi$ of type $\boldsymbol{\mathcal{T}}$ in $\Sigma$ through $t$. By Corollary 2.2.7, as $\pi$ contains a $3$-secant, $\pi$ is either a plane of type ${T^{\textnormal{odd}}}$, type ${T^{\boldsymbol{\triangle}}}$ or type ${T^{\bigstar}}$. Regardless of this type, $\pi$ always contains another $3$-secant $t^{\prime}$ such that $t\cap t^{\prime}\notin{\textnormal{supp}(c)}$.

Let $y$ be the number of type-$\boldsymbol{\mathcal{T}}$ planes in $\Sigma$ through $t^{\prime}$. Remark that such a plane intersects $\sigma$ in at most three points of ${\textnormal{supp}(c)}$. Indeed, should a $\boldsymbol{\mathcal{T}}$-typed plane in $\Sigma$ through $t^{\prime}$ intersect $\sigma$ in at least $4$, thus in at least $q-1$ points (Lemma 3.2.1), then one of the three points of $t^{\prime}\cap{\textnormal{supp}(c)}$ must lie on this intersection line (as $\pi$ is a plane of type $\boldsymbol{\mathcal{T}}$). But then $t^{\prime}\cap\sigma\in{\textnormal{supp}(c)}$, in contradiction with $t\cap t^{\prime}\notin{\textnormal{supp}(c)}$. In this way, we get

[TABLE]

which implies $y\leqslant\frac{1}{2}(q+7)$, as $q\geqslant 29$.

Thus we get that $t^{\prime}$ is contained in at least $q+1-\frac{1}{2}(q+7)=\frac{1}{2}(q-5)$ planes of type $\boldsymbol{\mathcal{O}}$ (all lying in $\Sigma$). As each $\boldsymbol{\mathcal{T}}$-typed plane in $\Sigma$ through $t^{\prime}$ contains at least $3q-3$ points of ${\textnormal{supp}(c)}$, we get

[TABLE]

As the above claim holds for all solids containing $\sigma$, we get

[TABLE]

One can easily check this implies ${\textnormal{wt}(c)}\geqslant B_{n,q}$ for all prime powers $q$, a contradiction. ∎

We can generalise the above lemma, which will prove its usefulness when using induction.

Lemma 3.2.9**.**

Suppose $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Let $\psi$ be a $k$-space, $2\leqslant k<n$, containing a $3$-secant $s$. Then ${\textnormal{wt}({c}_{|\psi})}\leqslant B_{k,q}$.

Proof.

By Lemma 3.2.8, we know that all planes in $\psi$ through $t$ contain at most $3q+1$ points of ${\textnormal{supp}(c)}$ (Corollary 2.2.7). This implies that ${\textnormal{wt}({c}_{|\psi})}\leqslant\theta_{k-2}(3q+1-3)+3\leqslant B_{k,q}$. ∎

Remark that the last inequality in the above proof is the reason why the bound $B_{n,q}$ differs in value for $q\in\{29,31,32\}$.

We can now present some properties about certain types of subspaces sharing a common $3$-secant.

Lemma 3.2.10**.**

Suppose $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Let $\Pi_{1}$ and $\Pi_{2}$ be two $k$-spaces, $2\leqslant k<n$, of type $T_{1},T_{2}\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}},{T^{\bigstar}}\}$, respectively, having a $3$-secant $s$ in common. Then at least one of the following holds:

1. (1)

$T_{1}={T^{\bigstar}}$. 2. (2)

$T_{2}={T^{\bigstar}}$. 3. (3)

$T_{1}=T_{2}$.

Furthermore, if $T_{1}=T_{2}$, then ${\textnormal{wt}({c}_{|\Pi_{1}})}={\textnormal{wt}({c}_{|\Pi_{2}})}$.

Proof.

In each subspace $\Pi_{i}$, choose a plane $\pi_{i}$ through $s$, disjoint to the vertex corresponding with the cone ${\textnormal{supp}({c}_{|\Pi_{i}})}$. By definition, $\pi_{i}$ is a plane of type $T_{i}$. Define $\Sigma={\langle\pi_{1},\pi_{2}\rangle}$.

Furthermore, let $P_{\alpha}$, $P_{\beta}$ and $P_{\gamma}$ be the points in $s\cap{\textnormal{supp}(c)}$ with corresponding non-zero values $\alpha$, $\beta$ and $\gamma$ in $c$. Let $l_{\alpha}^{(i)}$, $l_{\beta}^{(i)}$ and $l_{\gamma}^{(i)}$ be the unique long secants in $\pi_{i}$ through $P_{\alpha}$, $P_{\beta}$ and $P_{\gamma}$, respectively ($i=1,2$).

Case $1$: $T_{1}={T^{\textnormal{odd}}}$ and $T_{2}={T^{\boldsymbol{\triangle}}}$.

Suppose that $\pi$ is a plane in $\Sigma$ going through $l_{\alpha}^{(2)}$. Remark that $l_{\alpha}^{(2)}$ is a long secant, containing $q-1$ points having non-zero value $\alpha$, one point having value $\alpha+\beta$ and one point having value $\alpha+\gamma$. From this, we know that the plane $\pi$ cannot be

• •

a plane of type ${T_{0}}$, as $\alpha\neq 0$.

• •

a plane of type ${T_{q+1}}$, ${T_{2q}}$, ${T_{2q+1}}$ or ${T^{\bigstar}}$, else $\alpha+\beta=\alpha$ or $\alpha+\gamma=\alpha$.

• •

a plane of type ${T^{\textnormal{odd}}}$, as $l_{\alpha}^{(2)}$ contains at least three points with the same value $\alpha$.

• •

a plane of type ${T^{\boldsymbol{\triangle}}}$, unless ${\textnormal{wt}({c}_{|\pi})}={\textnormal{wt}({c}_{|\pi_{2}})}$. Indeed, $l_{\alpha}^{(2)}$ contains two points $l_{\alpha}^{(2)}\cap l_{\beta}^{(2)}$ and $l_{\alpha}^{(2)}\cap l_{\gamma}^{(2)}$ with corresponding values $\alpha+\beta$ and $\alpha+\gamma$, respectively, unambiguously fixing the weight of ${\textnormal{wt}({c}_{|\pi})}$.

However, $\pi$ can only be a plane of type ${T^{\boldsymbol{\triangle}}}$ in some cases. Suppose that $\pi$ is a plane of type ${T^{\boldsymbol{\triangle}}}$ and suppose that $\pi$ intersects $\pi_{1}$ in a $3$-secant $t$. One of the points of $t\cap{\textnormal{supp}(c)}$ is obviously $P_{\alpha}$, as this point belongs to both $l_{\alpha}^{(2)}$ and $\pi_{1}$. The other two points of $t\cap{\textnormal{supp}(c)}$ lie on $l_{\beta}^{(1)}$ and $l_{\gamma}^{(1)}$ and must have corresponding values $\beta$ and $\gamma$, as ${\textnormal{wt}({c}_{|\pi})}={\textnormal{wt}({c}_{|\pi_{2}})}$. As $\pi_{1}$ is a plane of type ${T^{\textnormal{odd}}}$, there are only two possibilities for $\pi$ to intersect $\pi_{1}$, namely when the $\beta$-valued point of $t$ lies on $l_{\beta}^{(1)}$ (then $\pi=\pi_{2}$), or when the $\beta$-valued point of $t$ lies on $l_{\gamma}^{(1)}$. Conclusion: of the at least $q-2$ planes through $l_{\alpha}^{(2)}$ in $\Sigma$, intersecting $\pi_{1}$ in a $3$-secant, at least $q-4$ of them cannot be a plane of type ${T^{\boldsymbol{\triangle}}}$, and thus must be planes of type $\boldsymbol{\mathcal{O}}$. In addition, the plane ${\langle l_{\alpha}^{(1)},l_{\alpha}^{(2)}\rangle}$ can never be a plane of type ${T^{\boldsymbol{\triangle}}}$ as well, as $l_{\alpha}^{(1)}$ contains many distinctly valued points. Thus, we find at least $q-3$ planes of type $\boldsymbol{\mathcal{O}}$ in $\Sigma$ through $l_{\alpha}^{(2)}$, each containing at least $\frac{1}{2}q(q-1)$ points of ${\textnormal{supp}(c)}$ (Lemma 3.2.1). The other planes in $\Sigma$ through $l_{\alpha}^{(2)}$, of which there are at most four, contain at least $3q-3$ points of ${\textnormal{supp}(c)}$. We get

[TABLE]

which is, if $n=3$, a direct contradiction or, if $n>3$, a contradiction with Lemma 3.2.9, as $\Sigma$ contains the $3$-secant $s$.

Case $2$: $T_{1}=T_{2}$.

Suppose, on the contrary, that ${\textnormal{wt}({c}_{|\Pi_{1}})}\neq{\textnormal{wt}({c}_{|\Pi_{2}})}$. W.l.o.g. we can assume that ${\textnormal{wt}({c}_{|\pi_{1}})}\neq{\textnormal{wt}({c}_{|\pi_{2}})}$ as well. Assume, in the first case, that $T_{1}\in\{{T^{\textnormal{odd}}},{T^{\bigstar}}\}$. By observing the types of these planes and by Proposition 2.2.4, ${\textnormal{wt}({c}_{|\pi_{1}})}\neq{\textnormal{wt}({c}_{|\pi_{2}})}$ implies that both $\alpha+\beta+\gamma=0$ and $\alpha+\beta+\gamma\neq 0$, a contradiction.

Now assume $T_{1}={T^{\boldsymbol{\triangle}}}$. Considering the plane $\pi_{i}$, we know that the lines $l_{\alpha}^{(i)}$, $l_{\beta}^{(i)}$ and $l_{\gamma}^{(i)}$ are not concurrent. As ${\textnormal{wt}({c}_{|\pi_{1}})}\neq{\textnormal{wt}({c}_{|\pi_{2}})}$, we know, without loss of generality, that the value of the point $l_{\alpha}^{(1)}\cap l_{\beta}^{(1)}$ is zero, while the value of the point $l_{\alpha}^{(2)}\cap l_{\beta}^{(2)}$ is not zero. This implies that $\alpha+\beta$ is both zero and non-zero, a contradiction. ∎

Lemma 3.2.11**.**

Suppose $q>3$ and let $\pi$ be a plane of type $T\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}}\}$. Then all planes $\sigma$ of type $\mathcal{T}$ intersecting $\pi$ in a long secant are planes of type $T$ as well. Moreover, ${\textnormal{wt}({c}_{|\sigma})}={\textnormal{wt}({c}_{|\pi})}$.

Proof.

Suppose the plane $\sigma$ is a plane of type $T_{\sigma}\in\mathcal{T}$; let $l$ be the long secant $\pi\cap\sigma$. As $T\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}}\}$, no $q$ points on $l$ have the same non-zero value in $c$. As a consequence, $T_{\sigma}\notin\{{T_{0}},{T_{q+1}},{T_{2q}},{T_{2q+1}},{T^{\bigstar}}\}$. If $T={T^{\textnormal{odd}}}$, we find at least $q$ points on $l$ having pairwise different values in $c$. If $T={T^{\boldsymbol{\triangle}}}$, we find at most $3$ different points on $l$ having pairwise different values. Hence, if $T_{\sigma}\neq T$, then $q\leqslant 3$, a contradiction. Furthermore, it is not hard to check that the set of values of points on $l$ fixes the weight of ${c}_{|\sigma}$. ∎

Lemma 3.2.12**.**

Suppose that $n=3$ and $3q^{2}-3q-3\leqslant{\textnormal{wt}(c)}\leqslant B_{3,q}$. Then a $3$-secant is never contained in $q+1$ planes of the same type $T\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}}\}$.

Proof.

Suppose, on the contrary, that $t$ is such a $3$-secant. Fix a plane $\pi$ through $t$. By Lemma 3.2.10, the weight of the code word $c$ is known, as we can count: {\textnormal{wt}(c)}=(q+1)\big{(}{\textnormal{wt}({c}_{|\pi})}-3\big{)}+3=(q+1){\textnormal{wt}({c}_{|\pi})}-3q.

Remark that, as $\pi$ is a plane of either type ${T^{\textnormal{odd}}}$ or ${T^{\boldsymbol{\triangle}}}$, we can always find a $1$- or $2$-secant $r$ in $\pi$ such that $t$ and $r$ intersect in a point $Q$ of ${\textnormal{supp}(c)}$. Indeed,

• •

if $\pi$ is a plane of type ${T^{\textnormal{odd}}}$, we can simply connect two points: a hole lying on a long secant in $\pi$, different from the intersection point of the three long secants in $\pi$, with a point of $t\cap{\textnormal{supp}(c)}$ on another long secant in $\pi$.

• •

if $\pi$ is plane of type ${T^{\boldsymbol{\triangle}}}$, we can connect a point lying on two long secants with the unique point of $t$ lying on the third long secant.

Let $\sigma$ be a plane through $r$, not equal to $\pi$. Choose a long secant $s$ in $\sigma$ through $Q$. This is possible since every plane of type $\mathcal{T}\setminus\{T_{0}\}$ obviously contains a long secant, and planes of type $\mathcal{O}$ contain long secants as well (cfr. Lemma 3.2.1). The plane ${\langle t,s\rangle}$ contains the $3$-secant $t$, thus this plane has to be of the same subtype as $\pi$. In particular, this means that ${\langle t,s\rangle}$ is a plane of type ${T^{\textnormal{odd}}}$ or ${T^{\boldsymbol{\triangle}}}$. However, by Lemma 3.2.11, the plane ${\langle t,s\rangle}$ then has to be of the same type as $\sigma$ as well, as they share the long secant $s$, unless $\sigma$ is a plane of type $\boldsymbol{\mathcal{O}}$.

Therefore, all planes $\sigma$ through $r$ satisfy either ${\textnormal{wt}({c}_{|\sigma})}={\textnormal{wt}({c}_{|\pi})}$ (if $\sigma$ is a plane of type $\mathcal{T}$), or ${\textnormal{wt}({c}_{|\sigma})}\geq\frac{1}{2}q(q-1)>{\textnormal{wt}({c}_{|\pi})}$ (if $\sigma$ is a plane of type $\mathcal{O}$, by Lemma 3.2.1). In both cases, this yields the following lower bound on ${\textnormal{wt}(c)}$:

[TABLE]

a contradiction. ∎

The following proposition is a consequence of the way code words of type $\boldsymbol{\mathcal{T}}$ are defined (Definition 3.1.3).

Proposition 3.2.13**.**

Suppose that $\Pi$ is a hyperplane of type $T\in\boldsymbol{\mathcal{T}}$, with $\kappa$ the $(n-4)$-dimensional vertex of ${\textnormal{supp}({c}_{|\Pi})}$. Suppose that $t$ is a $3$-secant contained in $\Pi$. Then $t$ is disjoint to $\kappa$ and all $q^{n-3}$ planes in $\Pi$ that contain $t$ but that are disjoint to $\kappa$ are planes of type $T$. The other $\theta_{n-4}$ planes in $\Pi$ through $t$ intersect $\kappa$ in a point and are all planes of type ${T^{\bigstar}}$.

Lemma 3.2.14**.**

Suppose that $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Then a $3$-secant is never contained in $\theta_{n-2}$ hyperplanes of the same type $T\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}}\}$.

Proof.

By Lemma 3.2.12, we can assume that $n>3$. Suppose that $t$ is a $3$-secant with the described property. Now define

[TABLE]

Fix an arbitrary $T$-typed plane $\pi_{0}\supseteq t$. As all hyperplanes through $t$ are of the same type $T$, all hyperplanes through $\pi_{0}$ have this property as well. Thus, the number of elements in $S$ with a fixed first argument $\pi_{0}$ equals $\theta_{n-3}$.

Fix an arbitrary $T$-typed hyperplane $\Pi_{0}\supseteq t$. By Proposition 3.2.13, the number of elements in $S$ with a fixed second argument $\Pi_{0}$ equals $q^{n-3}$ (the number of planes in $\Pi_{0}$ through $t$, disjoint to an $(n-4)$-subspace not intersecting $t$).

Let $x_{\pi}$ be the number of $T$-typed planes through $t$. By double counting, we get:

[TABLE]

As $x_{\pi}$ is known to be an integer, this is only valid when the fraction on the right is an integer. As $n>3$, this is never the case. ∎

Lemma 3.2.15**.**

Suppose that $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Let $\Pi_{1}$ and $\Pi_{2}$ be two hyperplanes of type ${T^{\bigstar}}$ and let $\mathcal{C}_{i}$ be the union of the three $(n-2)$-subspaces present in the linear combination ${c}_{|\Pi_{i}}$, thus intersecting in a common $(n-3)$-space $\kappa_{i}$ ($i=1,2$). Suppose that $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have an $(n-2)$-subspace in common. Then either

• •

$\kappa_{1}=\kappa_{2}$, or

• •

$n>3$* and there exists a solid $S$ containing a long secant that is only contained in planes in $S$ of type $\mathcal{T}$.*

Proof.

Let $\Sigma$ be the $(n-2)$-space that $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have in common. As $q>3$, $\Sigma$ must be one of the three subspaces present in the linear combination of both ${c}_{|\Pi_{1}}$ and ${c}_{|\Pi_{2}}$.

Suppose that $\kappa_{1}\neq\kappa_{2}$. As these are spaces of the same dimension, we can find a point $P_{1}\in\kappa_{1}\setminus\kappa_{2}$ and a point $P_{2}\in\kappa_{2}\setminus\kappa_{1}$; define $l={\langle P_{1},P_{2}\rangle}$. Remark that $l$ must be a $(q+1)$-secant to ${\textnormal{supp}(c)}$. This follows from the fact that every point of $l$ lies in $\Sigma\setminus\kappa_{i}$, for at least one choice of $i$. Looking in $\mathcal{C}_{i}$, we see that all points of $\Sigma\setminus\kappa_{i}$ lie in ${\textnormal{supp}(c)}$. Now take planes $\pi_{i}$ in $\Pi_{i}$, for $i=1,2$, through $l$, not contained in $\Sigma$. Due to this choice, it is clear that the plane $\pi_{i}$ will intersect each $(n-2)$-subspace of $\mathcal{C}_{i}$ in a line (through $P_{i}$). Define $S={\langle\pi_{1},\pi_{2}\rangle}$.

Choose a $(q+1)$-secant $s$ in $\pi_{1}$, different from $l$. As $P_{1}\neq P_{2}$, all planes in $S$ through $s$ (not equal to $\pi_{1}$) intersect $\pi_{2}$ in a $3$-secant and thus, by Lemma 3.2.8, are planes of type $\boldsymbol{\mathcal{T}}$. As $\pi_{1}$ is a plane of type $\boldsymbol{\mathcal{T}}$ as well, we know that all planes in $S$ through $s$ are planes of type $\boldsymbol{\mathcal{T}}$.

If $n=3$, we get that $(3q-6)(q+1)+3\leqslant{\textnormal{wt}(c)}={\textnormal{wt}({c}_{|S})}\leqslant q\cdot(2q)+(3q+1)=2q^{2}+3q+1$, which is only valid if $q<7$, contrary to the assumptions. ∎

The following theorem connects all previous results and proves Theorem 3.1.4.

Theorem 3.2.16**.**

Suppose $(3q-6)\theta_{n-2}+3\leqslant{\textnormal{wt}(c)}\leqslant B_{n,q}$. Then there exists a plane $\pi$ of type $T\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}},{T^{\bigstar}}\}$ and an $(n-3)$-space $\kappa$ such that $\pi\cap\kappa=\emptyset$ and

[TABLE]

Proof.

We will prove this by induction on $n$. When $n=2$, we can choose $\kappa=\emptyset$ and refer to Corollary 2.2.7. Now assume $n\geqslant 3$ and suppose the statement is true for $c$ restricted to any $k$-space, $2\leqslant k<n$. By Lemma 3.2.7, we can choose a $3$-secant $t$ with corresponding non-zero values $\alpha$, $\beta$ and $\gamma$. By the induction hypothesis and Lemma 3.2.9, each hyperplane through $t$ is a hyperplane of type $\boldsymbol{\mathcal{T}}$ and by Lemma 3.2.10, we know that there exist two specific types $T_{A}={T^{\bigstar}}$ and $T_{B}\in\{{T^{\textnormal{odd}}},{T^{\boldsymbol{\triangle}}},{T^{\bigstar}}\}$ such that all hyperplanes through $t$ are either of type $T_{A}$ or type $T_{B}$. Furthermore, by Lemma 3.2.14, we know that there exists at least one hyperplane through $t$ of type $T_{A}$; consider such a hyperplane $\Pi$. Remark that by Proposition 3.2.13, all planes through $t$ are planes of type $T_{A}$ or $T_{B}$ as well. We can now fix a certain plane $\pi$ as follows: if all planes through $t$ are planes of type $T_{A}$, choose $\pi$ to be an arbitrary plane through $t$, not contained in $\Pi$. Else, choose $\pi$ to be a plane through $t$ of type $T_{B}$. By Proposition 3.2.13, $\pi$ cannot be contained in $\Pi$.

Furthermore, we know that ${c}_{|\Pi}$ is a linear combination of three different $(n-2)$-subspaces of $\Pi$ through an $(n-3)$-space. Choose $\kappa$ to be this $(n-3)$-space. As all lines in $\Pi$, not disjoint to $\kappa$, are either [math]-, $1$-, $q$- or $(q+1)$-secants, we know that $\kappa$ must be disjoint to the $3$-secant $t$ and, furthermore, disjoint from the plane $\pi\supseteq t$, as that plane is not contained in $\Pi$.

For each point $P\in\kappa$, it is easy to see that $c(P)$ is equal to the sum of the values of the points on the $3$-secant $t$ (which is $\alpha+\beta+\gamma$).

As ${c}_{|\Pi}$ is a linear combination of three different $(n-2)$-spaces of $\Pi$ having the space $\kappa$ in common, we can choose one of those $(n-2)$-spaces $\Psi_{1}$; w.l.o.g. this space corresponds to the value $\alpha$. Choose an arbitrary $3$-secant $t_{1}$ in $\pi$ through the point $\Psi_{1}\cap\pi$, thus having corresponding non-zero values $\alpha$, $\beta_{1}$ and $\gamma_{1}$. By the induction hypothesis and Lemma 3.2.9, $\Pi_{1}={\langle\Psi_{1},t_{1}\rangle}$ is a hyperplane of type $\boldsymbol{\mathcal{T}}$. We claim that $\Pi_{1}$ is a hyperplane of type $T_{A}$. Indeed, let $\pi_{1}$ be a plane in $\Pi_{1}$ through $t_{1}$, thus intersecting $\Pi$ in a line of $\Psi_{1}$. Then this intersection line must be a $q$- or $(q+1)$-secant. By Lemma 3.2.8, $\pi_{1}$ has to be a plane of type $\boldsymbol{\mathcal{T}}$ and, more specifically, a plane of type $T_{A}$ (Lemma 3.2.11). As such, all planes in $\Pi_{1}$ through $t_{1}$ are planes of type $T_{A}$, thus $\Pi_{1}$ contains at least $\theta_{n-3}$ planes of type $T_{A}$ through a fixed $3$-secant ($t_{1}$). By Proposition 3.2.13, at least one of these planes is of the same type as $\Pi_{1}$, thus this hyperplane must be of type $T_{A}$.

Let $\kappa_{1}$ be the $(n-3)$-subspace of $\Pi_{1}$ in which the three hyperplanes of ${c}_{|\Pi_{1}}$ intersect. By Lemma 3.2.15, we know that $\kappa=\kappa_{1}$. In this way, it is easy to see that all points in $\Pi_{1}\setminus\kappa$ fulfil the desired property.

We can now repeat the above process by choosing another $(n-2)$-space $\Psi_{2}$ in one of the linear combinations of ${c}_{|\Pi}$ or ${c}_{|\Pi_{1}}$ and considering the span $\Pi_{2}={\langle\Psi_{2},t_{2}\rangle}$, with $t_{2}$ an arbitrary $3$-secant in $\pi$ through the point $\Psi_{2}\cap\pi$. All points in $\Pi_{2}\setminus\kappa$ will fulfil the desired property as well.

To conclude, if, for each point $P$ in $\pi$, there exists a sequence of $3$-secants $t_{1},t_{2},\dots,t_{m}\ni P$ in $\pi$ such that $t\cap t_{1}\in{\textnormal{supp}(c)}$ and $t_{i}\cap t_{i+1}\in{\textnormal{supp}(c)}$ for all $i\in\{1,2,\dots,m-1\}$, then this theorem is proven by consecutively repeating the above arguments. Unfortunately, not all points in $\pi$ satisfy this property. However, if a point $P\in\pi$ does not lie on such a (sequence of) $3$-secant(s), we can easily prove it lies on a [math]-, $1$- or $2$-secant $r$ in $\pi$ having $q$ points that do satisfy this first property. Thus, we already know the value of a lot of points in the hyperplane ${\langle\kappa,r\rangle}$, namely of precisely $|{\langle\kappa,r\rangle}|-|{\langle\kappa,P\rangle}|+|\kappa|=\theta_{n-1}-q^{n-2}$ points. Furthermore, ${\textnormal{wt}({c}_{|{\langle\kappa,r\rangle}})}\leqslant 2q^{n-2}+\theta_{n-3}+{\textnormal{wt}({c}_{|{\langle\kappa,P\rangle}})}-{\textnormal{wt}({c}_{|\kappa})}\leqslant 3q^{n-2}+\theta_{n-3}\leqslant B_{n-1,q}$. Thus, by the induction hypothesis, this hyperplane is a hyperplane of type $\boldsymbol{\mathcal{T}}$. It is easy to see that all points in ${\langle\kappa,P\rangle}$ must satisfy the property of the theorem. ∎

Acknowledgement. Special thanks to Maarten De Boeck for revising these results with great care and eye for detail.

Appendix A Further details to Lemma 3.2.1

Suppose $c\in C_{n-1}(n,q)$, with $q\geqslant 7$, $q\notin\{8,9,16,25,27,49\}$, and assume that ${\textnormal{wt}(c)}\leqslant D_{n,q}$, with

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Remark that $B_{n,q}<D_{n,q}$ if $q\in\{29,31,32\}$ and $B_{n,q}=D_{n,q}$ for all other considered values of $q$, so it suffices to check the details of the lemma for this bound $D_{n,q}$. We will prove a contradiction using the following two inequalities:

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Define $\boldsymbol{W}:={\textnormal{wt}(c)}$. Below, we will sketch the details when $q>17$, $q\notin\{25,27,49\}$. The other two cases are completely analogous.

Combining the two equations in (4), knowing that $A_{q}=4q-21$, gives rise to the following inequality:

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The above inequality is of the form $0\geqslant a\boldsymbol{W}^{2}+b\boldsymbol{W}+c$, with $a\geqslant 0$, implying that $\boldsymbol{W}\geqslant\frac{-b-\sqrt{D}}{2a}$ with $D=\sqrt{b^{2}-4ac}$. One can check that

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Keeping in mind that $q\geqslant 23$, we can raise the right-hand side and obtain

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On the other hand, we have that D^{2}\geqslant\big{(}-b-2a(4q-\sqrt{8q}-\frac{33}{2})\big{)}^{2}, which implies

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Keeping in mind that $q\geqslant 23$, we can lower the right-hand side and obtain

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Combining (5) and (6), we obtain

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resulting in

[TABLE]

a contradiction.