On the enumeration and classification of σ-LCD codes over finite commutative chain rings

Let R-e denote a finite commutative chain ring with the maximal ideal (u) and the nilpotency index e, and let (R) over bar (e) = R-e/(u) be the residue field of R-e. Let sigma(0) be an automorphism of R-e, and let (sigma) over bar (0) be the corresponding automorphism of the residue field (R) over bar (e) of R-e, defined as sigma(0)(a + (u)) = sigma(0)(a) + (u) for all a + (u) is an element of(R) over bar (e). Let sigma be an automorphism of R-e(n) corresponding to the automorphism sigma(0) of R-e, defined as sigma (v(1), v(2), . . . , v(n)) = (sigma(0)(v(1)), sigma(0)(v(2)), . . . , sigma(0)(v(n))) for all (v(1), v(2), ... , v(n)) is an element of R-e(n). In this paper, we obtain explicit enumeration formulae for all sigma-LCD codes of an arbitrary length over the chain ring R-e when (sigma) over bar (2)(0) is the identity automorphism of (R) over bar (e). With the help of these enumeration formulae and by applying the classification algorithm, we classify all Euclidean LCD codes of lengths 2, 3, 4 and 5 over the chain ring F-2[u]/(u(2)) and of lengths 2, 3 and 4 over the chain ring F-3[u]/(u(2)), and all sigma-LCD codes of lengths 2, 3 and 4 over the chain ring F-4[u]/(u(2)), where sigma(0) is an automorphism of F-4[u]/((u2)) such that the corresponding automorphism sigma 0 of the residue field F4 has order 2. Besides this, we show that the class of sigma-LCD codes over Re is asymptotically good, and that every free linear [n, k, d]-code over Re is equivalent to a sigma-LCD [n, k, d]-code over Re when |(R) over bar (e)| > 4. We also explicitly determine all inequivalent sigma-LCD [n, 1, d]-codes and [n, n - 1, d]-codes over R-e for 1 <= d <= n. (C) 2022 Elsevier B.V. All rights reserved.