Non-existence of some 4-dimensional Griesmer codes over finite fields

Year 2018, Volume: 5 Issue: 2, 101 - 116, 28.05.2018

Kazuki Kumegawa Tatsuya Maruta

https://doi.org/10.13069/jacodesmath.427968

Cited By: 3

Abstract

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$;
$d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil
d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for
$2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$,
$2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and
$q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$
and that $n_q(4,d) \ge g_q(4,d)+1$ for
$2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and
$2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$,
where $n_q(4,d)$ denotes the minimum length $n$ for which
an $[n,4,d]_q$ code exists.

Keywords

Optimal linear codes, Griesmer bound, Arcs in PG$(rq)$

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