In this paper we generalize the construction of Griesmer codes of Belov type. 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, givingwhere n q (k, d) is the minimum length n for which an [n, k, d] q code exists.
a b s t r a c tWe construct a lot of new [n, 5, d] 5 codes close to the Griesmer bound and prove the nonexistence of some Griesmer codes to determine the exact value of n 5 (5, d) or to improve the known upper bound on n 5 (5, d), where n q (k, d) is the minimum length n for which an [n, k, d] q code exists. We also give the updated table for n 5 (5, d) for all d except some known cases.
Abstract. It is well-known that there exists a constant-weight [sθ k−1 , k, sq k−1 ]q code for any positive integer s, which is an s-fold simplex code, where θ j = (q j+1 − 1)/(q − 1). This gives an upper bound nq(k, sq) is the minimum length n for which an [n, k, d]q code exists. We construct a two-weight [sθ k−1 + 1, k, sq k−1 ]q code for 1 ≤ s ≤ k − 3, which gives a better upper bound nq(k, sqAs another application, we prove that nq(5, d) = 4 i=0 d/q i for q 4 + 1 ≤ d ≤ q 4 + q for any prime power q.
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