Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2017; 54(3): 731-736

Online first article March 20, 2017      Printed May 31, 2017

https://doi.org/10.4134/BKMS.b151011

Copyright © The Korean Mathematical Society.

A construction of two-weight codes and its applications

Eun Ju Cheon, Yuuki Kageyama, Seon Jeong Kim, Namyong Lee, and Tatsuya Maruta

Gyeongsang National University, Osaka Prefecture University, Gyeongsang National University, Minnesota State University, Osaka Prefecture University

Abstract

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 $q$.

Keywords: linear code, two-weight code, length optimal code, Griesmer bound, projective space

MSC numbers: 94B27, 94B05, 51E20, 05B25