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

Abstract

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space $\text {PG}(n,q)$. Recently, Polverino and Zullo (J Comb Theory Ser A 158:1–11, 2018) 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 prove that all code words of weight at most $\big (4q-{\mathcal {O}} (\sqrt{q})\big )q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a common $(n-3)$-space. This extends previous results for large values of q.

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 ${\text{ wt }(c)}\leqslant E_{n,q}$, with

$$\begin{aligned} E_{n,q}= & {} {\left\{ \begin{array}{ll} \Big (3q-\sqrt{6q}-\frac{1}{2}\Big )q^{n-2}\;&{}\text{ if } q\in \{7,11,13,17\}\text{, }\\ \Big (3q-\sqrt{6q}+\frac{9}{2}\Big )q^{n-2}\;&{}\text{ if } q\in \{19,121\}\text{, }\\ \Big (4q-\sqrt{8q}-\frac{33}{2}\Big )q^{n-2}\;&{}\text{ otherwise; }\\ \end{array}\right. }\\ A_q= & {} {\left\{ \begin{array}{ll} 3q-3\;&{}\text{ if } q\in \{7,11,13,17\}\text{, }\\ 3q+2\;&{}\text{ if } q\in \{19,121\}\text{, }\\ 4q-21\;&{}\text{ otherwise. }\\ \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} {\text{ wt }(c)}\geqslant \bigg (\frac{1}{2}j(j+1)-j\bigg )\theta _{n-2}+j\qquad \text{ and }\qquad j\geqslant \frac{A_q\theta _{n-2}-{\text{ wt }(c)}}{\theta _{n-2}-1}\text{. } \end{aligned}$$

(6)

Define $\varvec{W}:={\text{ wt }(c)}$. Below, we will sketch the details when $q>19$, and $q \not = 121$. The other two cases are completely analogous.

Combining the two equations in (6), together with $A_q=4q-21$, gives rise to the following inequality:

$$\begin{aligned} 0\geqslant (&q^{n+1}-2q^n+q^{n-1}-q^2+2q-1)\varvec{W}^2\\ -&(8q^{2n}-49q^{2n-1}+41q^{2n-2}-17q^{n+1}+100q^n-83q^{n-1}+9q^2-51q+42)\varvec{W}\\ +&16q^{3n-1}-172q^{3n-2}+462q^{3n-3}-36q^{2n}+441q^{2n-1}-1323q^{2n-2}\\ -&8q^{n+2}+82q^{n+1}-458q^n+1302q^{n-1}+8q^3-62q^2+189q-441\text{. } \end{aligned}$$

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

$$\begin{aligned} D&=32q^{4n-1}-231q^{4n-2}+366q^{4n-3}-167q^{4n-4}\\&\quad -64q^{3n+1}+398q^{3n}-270q^{3n-1}-398q^{3n-2}+334q^{3n-3}\\&\quad +32q^{2n+3}-103q^{2n+2}-526q^{2n+1}+1066q^{2n}-302q^{2n-1}-167q^{2n-2}\\&\quad -64q^{n+4}+398q^{n+3}-270q^{n+2}-398q^{n+1}+334q^n\\&\quad +32q^5-231q^4+366q^3-167q^2\text{. } \end{aligned}$$

Since $q\geqslant 23$, we can find the following upper bound on the right-hand side:

$$\begin{aligned} D\leqslant 32q^{4n-1}-231q^{4n-2}+398q^{4n-3}-46q^{3n+1}\text{. } \end{aligned}$$

(7)

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

$$\begin{aligned} D&\geqslant 32q^{4n-1}-128q^{4n-2}-264\sqrt{2q}\cdot q^{4n-3}+192q^{4n-3}+792\sqrt{2q}\cdot q^{4n-4}+961q^{4n-4}\\&\quad -792\sqrt{2q}\cdot q^{4n-5}-2146q^{4n-5}+264\sqrt{2q}\cdot q^{4n-6}+1089q^{4n-6}\\&\quad -72\sqrt{2q}\cdot q^{3n}-64q^{3n}+552\sqrt{2q}\cdot q^{3n-1}+850q^{3n-1}-696\sqrt{2q}\cdot q^{3n-2}\\&\quad -4344q^{3n-2}-504\sqrt{2q}\cdot q^{3n-3}+4216q^{3n-3}+1248\sqrt{2q}\cdot q^{3n-4}+1520q^{3n-4}\\&\quad -528\sqrt{2q}\cdot q^{3n-5}-2178q^{3n-5}+81q^{2n+2}+144\sqrt{2q}\cdot q^{2n+1}-886q^{2n+1}\\&\quad -1104\sqrt{2q}\cdot q^{2n}+2041q^{2n}+2184\sqrt{2q}\cdot q^{2n-1}+3828q^{2n-1}-1368\sqrt{2q}\cdot q^{2n-2}\\&\quad -9551q^{2n-2}-120\sqrt{2q}\cdot q^{2n-3}+3398q^{2n-3}+264\sqrt{2q}\cdot q^{2n-4}+1089q^{2n-4}\\&\quad -162q^{n+3}-72\sqrt{2q}\cdot q^{n+2}+1836q^{n+2}+552\sqrt{2q}\cdot q^{n+1}-6120q^{n+1}\\&\quad -1224\sqrt{2q}\cdot q^n+4608q^n+1080\sqrt{2q}\cdot q^{n-1}+2610q^{n-1}-336\sqrt{2q}\cdot q^{n-2}\\&\quad -2772q^{n-2}+81q^4-918q^3+3357q^2-4284q+1764\text{. } \end{aligned}$$

Since $q\geqslant 23$, we can find the following lower bound on the right-hand side:

$$\begin{aligned} D\geqslant 32q^{4n-1}-206q^{4n-2}-72\sqrt{2q}\cdot q^{3n}-64q^{3n}\text{. } \end{aligned}$$

(8)

Combining (7) and (8) yields

$$\begin{aligned}&32q^{4n-1}-231q^{4n-2}+398q^{4n-3}-46q^{3n+1}\geqslant D\geqslant 32q^{4n-1}-206q^{4n-2}\\&\quad -72\sqrt{2q}\cdot q^{3n}-64q^{3n}\text{, } \end{aligned}$$

resulting in

$$\begin{aligned}&0\geqslant 25q^{4n-2}-398q^{4n-3}+46q^{3n+1}-72\sqrt{2q}\cdot q^{3n}-64q^{3n}\\ \Longrightarrow \quad&0\geqslant 25q^{4n-2}-398q^{4n-3}\\ \Longrightarrow \quad&\frac{398}{25}\geqslant q\text{, } \end{aligned}$$

a contradiction.