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.
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
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
2008; 45(3): 419-425
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2004; 41(1): 19-25
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