Linear Codes from Projective Linear Anticodes Revisited

Abstract

An anticode $\mathbf{C}\subset {\mathbf{F}}_q^n$ with the diameter $\hat{I}´$ is a code in ${\mathbf{F}}_q^n$ such that the distance between any two distinct codewords in $\mathbf{C}$ is at most $\hat{I}´$. The famous Erdös-Kleitman bound for a binary anticode $\mathbf{C}$ of the length $n$ and the diameter $\hat{I}´$ asserts that $$|\mathbf{C}|\le \hat{I}{\pounds }_{i=0}^{\frac{\hat{I}}{´}2}\left(\genfrac{}{}{0ex}{}{n}{i}\right).$$ In this paper, we give an antiGriesmer bound for $q$-ary projective linear anticodes, which is stronger than the above Erdös-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 $l\ge 3$.

38 pages, submitted