The hull of two classical propagation rules and their applications

TL;DR

This paper investigates the hull dimensions of classical propagation rules in coding theory, providing new criteria for code properties and constructing numerous optimal or near-optimal codes with specific hull characteristics.

Contribution

It introduces new criteria for self-duality, self-orthogonality, and LCD properties of codes derived from classical propagation rules, and constructs many new codes with prescribed hull dimensions.

Findings

Constructed new binary, ternary, and quaternary codes with specific hull properties.

Provided criteria for codes to be self-dual, self-orthogonal, or LCD.

Improved lower bounds on the minimum distance of LCD codes.

Abstract

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As an application, we construct some linear codes with prescribed hull dimensions, many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, and quaternary Hermitian FSD LCD codes. Some new even-like, odd-like, Euclidean and Hermitian self-orthogonal codes are also obtained. Many of {these} codes are also (almost) optimal according to the Database maintained by Markus Grassl. Our methods contribute positively to improve the lower bounds on the minimum distance of known LCD codes.