Linear Codes from Projective Linear Anticodes Revisited

An anticode C Fqⁿ with the diameter is a code in Fqⁿ such that the distance between any two distinct codewords in C is at most. The famous Erd\"os-Kleitman bound for a binary anticode C of the length n and the diameter asserts that | C| ₈=₀^{2} n i. In this paper, we give an antiGriesmer bound for q-ary projective linear anticodes, which is stronger than the above Erd\"os-Kleitman bound for binary anticodes. The antiGriesmer bound is a lower bound on diameters of projective linear anticodes. From some known projective linear anticodes, we construct some linear codes with optimal or near optimal minimum distances. A complementary theorem constructing infinitely many new projective linear (t+1) -weight code from a known t-weight linear code is presented. Then many new optimal or almost optimal few-weight linear codes are given and their weight distributions are determined. As a by-product, we also construct several infinite families of three-weight binary linear codes, which lead to l-strongly regular graphs for each odd integer l 3.