Hulls of free linear codes over a non-unital ring

This paper investigates the hull codes of free linear codes over a non-unital ring $ E = \langle \kappa, \tau \mid 2 \kappa = 2 \tau = 0, \; \kappa^2 = \kappa, \; \tau^2 = \tau, \; \kappa \tau = \kappa, \; \tau \kappa = \tau \rangle $. Initially, we examine the residue and torsion codes of various hulls of $ E $-linear codes and obtain an explicit form of the generator matrix of the hull of a free $ E $-linear code. Then, we propose four build-up construction methods to construct codes with a larger length and hull-rank from codes with a smaller length and hull-rank. Some illustrative examples are also given to support our build-up construction methods. Subsequently, we study the permutation equivalence of two free $ E $-linear codes and discuss the hull-variation problem. As an application, we classify optimal free $ E $-linear codes for lengths up to 8.