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