A construction of two-weight codes and its applications
Bull. Korean Math. Soc. 2017 Vol. 54, No. 3, 731-736
Published online March 20, 2017
Printed May 31, 2017
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
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