Linear Codes from Projective Linear Anticodes Revisited

TL;DR

This paper introduces a new antiGriesmer bound for projective linear anticodes, leading to the construction of optimal or near-optimal linear codes with specific weight properties and applications to strongly regular graphs.

Contribution

It establishes a stronger antiGriesmer bound for q-ary projective linear anticodes and constructs new linear codes with optimal or near-optimal parameters and specific weight distributions.

Findings

Derived a stronger antiGriesmer bound for projective linear anticodes.

Constructed new linear codes with optimal or near-optimal minimum distances.

Produced infinite families of three-weight binary linear codes related to strongly regular graphs.

Abstract

An anticode ${\bf C} \subset {\bf F}_q^n$ with the diameter $\delta$ is a code in ${\bf F}_q^n$ such that the distance between any two distinct codewords in ${\bf C}$ is at most $\delta$. The famous Erd\"{o}s-Kleitman bound for a binary anticode ${\bf C}$ of the length $n$ and the diameter $\delta$ asserts that $$|{\bf C}| \leq \Sigma_{i=0}^{\frac{\delta}{2}} \displaystyle{n \choose i}.$$ In this paper, we give an antiGriesmer bound for $q$-ary projective linear anticodes, which is stronger than the above Erd\"{o}s-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 $l \geq 3$.