A construction of two-weight codes and its applications. (English) Zbl 1370.94590

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

Bull. Korean Math. Soc. 54, No. 3, 731-736 (2017).

Summary: 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 \(n_q(k, s q^{k-1}+d) \leq s \theta_{k-1} +1 + n_q(k-1,d)\) 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 \(q\).

MSC:

94B27	Geometric methods (including applications of algebraic geometry) applied to coding theory
94B05	Linear codes, general
51E20	Combinatorial structures in finite projective spaces
05B25	Combinatorial aspects of finite geometries

Keywords:

linear code; two-weight code; length optimal code; Griesmer bound; projective space