A further study on the mass formula for linear codes with prescribed hull dimension

TL;DR

This paper develops a general technique to derive mass formulas for linear codes with prescribed hull dimensions, using group actions on Hermitian and symplectic LCD codes, and provides asymptotic analysis.

Contribution

It introduces a novel method to compute mass formulas for codes with specific hull dimensions via group actions, extending previous results.

Findings

All Hermitian and symplectic LCD codes form a unique orbit under group action.

Derived explicit formulas for the size of these orbits.

Established asymptotic behavior of the mass formulas.

Abstract

Finding a mass formula for a given class of linear codes is a fundamental problem in combinatorics and coding theory. In this paper, we consider the action of the unitary (resp. symplectic) group on the set of all Hermitian (resp. symplectic) linear complementary dual (LCD) codes, prove that all Hermitian (resp. symplectic) LCD codes are on a unique orbit under this action, and determine the formula for the size of the orbit. Based on this, we develop a general technique to obtain a closed mass formula for linear codes with prescribed Hermitian (resp. symplectic) hull dimension, and further obtain some asymptotic results.