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ON THE CONSTRUCTION OF OPTIMAL LINEAR CODES OF DIMENSION FOUR
Atsuya Kato,Tatsuya Maruta,Keita Nomura 대한수학회 2023 대한수학회보 Vol.60 No.5
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 $d$.
A construction of two-weight codes and its applications
전은주,Yuuki Kageyama,김선정,이남영,Tatsuya Maruta 대한수학회 2017 대한수학회보 Vol.54 No.3
It is well-known that there exists a constant-weight $[s \theta_{k-1},k, $ $ sq^{k-1}]_q$ code for any positive integer $s$, which is an $s$-fold simplex code, where $\theta_{j}=(q^{j+1}-1)/(q-1)$. This gives an upper bound $n_q(k, s q^{k-1}+d) \le s \theta_{k-1} + n_q(k,d)$ for any positive integer $d$, where $n_q(k,d)$ is the minimum length $n$ for which an $[n,k,d]_q$ code exists. We construct a two-weight $[s \theta_{k-1}+1,k, s q^{k-1}]_q$ code for $1 \le s \le k-3$, which gives a better upper bound $n_q(k, s q^{k-1}+d) \le s \theta_{k-1} +1 + n_q(k-1,d)$ for $1 \le d \le q^s$. As another application, we prove that $n_q(5,d)=\sum_{i=0}^{4}{\left\lceil{{d}/{q^i}}\right\rceil}$ for $q^{4}+1 \le d \le q^4+q$ for any prime power $q$.
A CONSTRUCTION OF TWO-WEIGHT CODES AND ITS APPLICATIONS
Cheon, Eun Ju,Kageyama, Yuuki,Kim, Seon Jeong,Lee, Namyong,Maruta, Tatsuya Korean Mathematical Society 2017 대한수학회보 Vol.54 No.3
It is well-known that there exists a constant-weight $[s{\theta}_{k-1},k,sq^{k-1}]_q$ code for any positive integer s, which is an s-fold simplex code, where ${\theta}_j=(q^{j+1}-1)/(q-1)$. This gives an upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+n_q(k,d)$ for any positive integer d, where $n_q(k,d)$ is the minimum length n for which an $[n,k,d]_q$ code exists. We construct a two-weight $[s{\theta}_{k-1}+1,k,sq^{k-1}]_q$ code for $1{\leq}s{\leq}k-3$, which gives a better upper bound $n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+1+n_q(k-1,d)$ for $1{\leq}d{\leq}q^s$. As another application, we prove that $n_q(5,d)={\sum_{i=0}^{4}}{\lceil}d/q^i{\rceil}$ for $q^4+1{\leq}d{\leq}q^4+q$ for any prime power q.