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ON THE CONSTRUCTION OF OPTIMAL LINEAR CODES OF DIMENSION FOUR

Atsuya Kato|Tatsuya Maruta|Keita Nomura
대한수학회 (The Korean Mathematical Society)|2023.09|대한수학회보 (Bulletin of the KMS)|제60권 제5호|pp. 1237-1252|DOI 10.4134/BKMS.b220613

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참고문헌

키워드

Linear codesdivisible codesprojective dualgeometric method
주제분류수학

초록

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 $\bF_q$, 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 $\bF_q$ with $q=7,8$, which determine $n_7(4,d)$ for $31$ values of $d$ and $n_8(4,d)$ for $40$ values of $d$.

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 $\bF_q$, 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 $\bF_q$ with $q=7,8$, which determine $n_7(4,d)$ for $31$ values of $d$ and $n_8(4,d)$ for $40$ values of $d$.

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