Bull. Korean Math. Soc. 2023; 60(5): 1237-1252
Online first article July 19, 2023 Printed September 30, 2023
https://doi.org/10.4134/BKMS.b220613
Copyright © The Korean Mathematical Society.
ATSUYA KATO, TATSUYA MARUTA, KEITA NOMURA
Osaka Prefecture University; Osaka Metropolitan University; Osaka Prefecture University
A fundamental problem in coding theory is to find $n_q(k,d)$, the minimum length $n$ for which an $[n,k,d]_q$ code exists. We show that some $q$-divisible optimal linear codes of dimension $4$ over $\mbox{$\mathbb{F}$}_q$, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG$(3,q)$. We also construct some new linear codes over $\mbox{$\mathbb{F}$}_q$ with $q=7,8$, which determine $n_7(4,d)$ for $31$ values of $d$ and $n_8(4,d)$ for $40$ values of $d$.
Keywords: Linear codes, divisible codes, projective dual, geometric method
MSC numbers: Primary 94B27, 51E20
Supported by: The second author is partially supported by JSPS KAKENHI Grant Number 20K03722.
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