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

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
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