Linear Codes from Projective Linear Anticodes Revisited

TL;DR

The paper revisits linear codes arising from projective linear anticodes, introducing an antiGriesmer bound that tightens diameter constraints and enabling a complementary construction that converts a $t$-weight projective code into infinitely many $(t+1)$-weight codes with explicit weight distributions. It establishes a general framework (Theorem 3.1 and the complementary theorem) linking anticodes to projective codes and yields numerous infinite families of optimal, almost optimal, and minimal codes, including complementary MDS and Reed-Solomon variants. A broad array of few-weight codes is produced, with many attaining or approaching Griesmer-type optimality, and practical by-products include $l$-strongly walk-regular graphs derived from binary three-weight codes. The results expand the catalog of explicit, well-characterized codes and offer tools for constructing highly structured minimal and few-weight codes with applications in combinatorics and graph theory.

Abstract

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 $l \geq 3$.

Theorems & Definitions (28)

• Theorem 1.1: AntiGriesmer bound
• Theorem 1.2
• Theorem 1.3: Complementary theorem
• Theorem 1.4: Complementary Reed-Solomon codes
• Proposition 2.1
• Theorem 2.1
• Corollary 2.1
• Corollary 3.1: Complementary theorem
• Theorem 5.1
• Theorem 5.2