Linear Codes from Projective Linear Anticodes Revisited
H Chen, Conghui Xie · arXiv (Cornell University) · 2024
An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $δ$ is a code in ${\bf F}_q^n$ such that the distance between any two distinct codewords in ${\bf C}$ is at most $δ$. The famous Erdös-Kleitman bound for a binary anticode ${\bf C}$ of the length $n$ and the diameter $δ$ asserts that $$|{\bf C}| \leq Σ_{i=0}^{\fracδ{2}} \displaystyle{n \choose i}.$$ 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 $l \geq 3$.