Core Concepts
The paper characterizes the algebraic structure of 2-quasi λ-constacyclic codes over finite fields and their Galois self-dual properties. It also investigates the asymptotic goodness of Hermitian and Euclidean self-dual 2-quasi λ-constacyclic codes.
Abstract
The paper studies 2-quasi λ-constacyclic codes over finite fields F with cardinality q = pℓ, where p is a prime and ℓ is a positive integer. It characterizes the algebraic structure of these codes and their Galois self-dual properties:
It shows that any 2-quasi λ-constacyclic code C over F can be written as (C1 × C2) ⊕ Cb,bg, where C1, C2, Cb are ideals of the quotient algebra Rλ = F[X]/(Xn - λ) and g is a unit in Cb.
It provides necessary and sufficient conditions for a 2-quasi λ-constacyclic code to be Galois self-dual, depending on whether λ1+ph = 1 or not.
It proves that if λ1+ph ≠ 1, then the Galois self-dual 2-quasi λ-constacyclic codes are asymptotically bad.
When ℓ is even and λ1+pℓ/2 = 1, it shows that the Hermitian self-dual 2-quasi λ-constacyclic codes are asymptotically good.
When pℓ ≠ 3 (mod 4) and λ2 = 1, it proves that the Euclidean self-dual 2-quasi λ-constacyclic codes are asymptotically good.
The paper introduces a new operator "∗" on the quotient ring Rλ, which becomes a useful technique for studying the Galois duality property of λ-constacyclic codes.