Abstract
In this paper we generalize the construction of Griesmer codes of Belov type to construct \([g_q(k,d)+t,k,d]_q\) codes with an integer \(t \ge 1\), where \(g_q(k,d)=\sum _{i=0}^{k-1} \left\lceil d/q^i \right\rceil \). This leads to the construction of several codes of length \(g_q(k,d)+1\), many of which are optimal. We also construct a q-divisible \([q^2+q,5,q^2-q]_q\) code through projective geometry. As a projective dual of the code, we construct optimal codes, giving \(n_q(5,d)=g_q(5,d)+1\) for \(q^4-q^3-q^2+1 \le d \le q^4-q^3-2q\), \(q \ge 3\), where \(n_q(k,d)\) is the minimum length n for which an \([n,k,d]_q\) code exists.
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We would like to thank the referees for their careful reading and helpful comments. The second author was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 24540138.
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Communicated by I. Landjev.
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Kageyama, Y., Maruta, T. On the geometric constructions of optimal linear codes. Des. Codes Cryptogr. 81, 469–480 (2016). https://doi.org/10.1007/s10623-015-0167-2
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DOI: https://doi.org/10.1007/s10623-015-0167-2