Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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.

On the construction of optimal linear codes of dimension four

ATSUYA KATO, TATSUYA MARUTA, KEITA NOMURA

Osaka Prefecture University; Osaka Metropolitan University; Osaka Prefecture University

Abstract

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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