Galois Self-Dual Constacyclic Codes

1 Introduction

• Earlier work was mostly in the semisimple case, where the paper notes fewer results on Hermitian self-dual constacyclic codes.The introduction also situates the work within research on constacyclic-code duality and self-duality.Source: §1 · pp. 1–2
• Galois inner products generalize the Euclidean and Hermitian inner products, and the paper defines corresponding Galois dual and self-dual constacyclic codes.The usual Euclidean inner product occurs at h = 0; the Hermitian inner product occurs when e is even and h = e/2.Source: §1 · p. 2
• The paper characterizes constacyclic codes with q-coset functions, including when p divides the code length.In that setting, constacyclic codes are no longer characterized by sets of zeros.Source: §1 · p. 2
• It defines isometries between constacyclic codes in the general setting and uses them to characterize Galois dual codes by q-coset functions.The characterization also yields results for dual and Hermitian-dual codes.Source: §1 · p. 2
• The paper obtains necessary and sufficient existence conditions for Galois self-dual and isometrically Galois self-dual constacyclic codes.These conditions cover the isometrically self-dual and isometrically Hermitian self-dual cases; self-dual and Hermitian self-dual existence results follow as consequences.Source: §1 · pp. 2–3

2 Constacyclic codes and q-coset functions

• The quotient ring Rn,λ = Fq[X]/⟨Xn − λ⟩ is formed over Fq, and C is a λ-constacyclic code of length n over Fq when C is an ideal of that ring.Source: §2 · p. 3
• q-orbits on 1 + rZ_(n′r) form q-cosets, and the roots of X^n − λ can occur with multiplicity p^νp(n).The section constructs these orbits from permutations of residue classes; the roots in the relevant classes have the stated multiplicity.Source: §2 · p. 4
• A q-coset function is constant on each q-orbit and takes values in [0, p^νp(n)].The complement function is defined by subtracting each value from p^νp(n).Source: §2 · p. 5
• Each λ-constacyclic code has a unique q-coset function; its check polynomial is f_ϕ(X), its generator polynomial is f_(complement ϕ)(X), and its annihilator is the code for the complement function.The section denotes the code associated with ϕ by C_ϕ.Source: §2 · pp. 5–6

3 Ring isometries

• For s coprime to n′r, M_s is a ring isomorphism between the constacyclic quotient rings and is generally semilinear over F_q.Its semilinearity is given by the Galois automorphism γ_(p^νp(s)).Source: §3 · pp. 6–7
• The ring isometries are defined even when p divides n, unlike the cited isometries restricted to the semisimple case.The paper describes these maps as semi-linear algebra isomorphisms in general.Source: §3 · p. 7
• The section establishes when two parameters give the same ring isometry and when they induce the same action on q-coset functions.The stated equivalences concern equality of M_s maps and equality of sϕ for every q-coset function.Source: §3 · pp. 7, 9
• The map M_s sends a λ-constacyclic code C_ϕ to the λ^s-constacyclic code C_(sϕ).The action on q-coset functions is defined by transporting their values under the corresponding residue-class map.Source: §3 · pp. 8–9

• The paper calls the Galois dual of a code its ph-dual code.Source: §4 · p. 9
• The p^h Galois inner product is linear in its first variable and γ^ph-linear in its second; h=0 gives the Euclidean product, and h=e/2 gives the Hermitian product when e is even.Source: §4 · pp. 9–10
• The section characterizes Galois dual codes of constacyclic codes by q-coset functions.Source: §4 · p. 10
• Theorem 4.4 is identified as the main result of the q-coset-function characterization.Source: §4 · pp. 10–11

• The q-coset-function orbit pattern alternates values on even s-orbits; on odd s-orbits, all values are 1.Source: §5 · p. 12
• Cϕ is isometrically ph-self-dual exactly when some s coprime to n′r, with s ≡ 1 (mod r), satisfies sϕ = ϕ̄.Source: §5 · p. 13
• The existence criterion in Lemma 5.4 allows either p = 2 with ν2(n) ≥ 1 or even length for every s-orbit.Source: §5 · p. 13
• For odd q, the paper equates the existence of duadic λ′-constacyclic codes with the even-s-orbit condition used in the isometric self-duality criterion.Source: §5 · p. 14
• Theorem 5.6 gives necessary and sufficient conditions for existence, including p odd, ν2(r) = 1, and min{ν2(q + 1), ν2(n′)} ≥ 2.Source: §5 · p. 14

6 Galois self-dual constacyclic codes

• The section characterizes Galois self-dual codes using q-coset functions: the relevant function must satisfy −p^hϕ = ¯ϕ under the stated divisibility condition.The theorem equates this function condition with the corresponding self-duality condition on the code.Source: §6 · p. 15
• The even-orbit criterion requires n′ and r both even and one of three conditions involving p modulo 4, the parities of e and h, and ν2(n′r).The three cases distinguish p ≡ 1 (mod 4) from p ≡ −1 (mod 4), with the latter depending on e, h, and the valuation threshold.Source: §6 · p. 16
• For p ≡ −1 (mod 4), another existence case requires n′ and r even, at least one of e and h odd, and ν2(n′r) > v.Here p = −1 + 2^v u with u odd and v ≥ 2.Source: §6 · p. 18
• The section derives existence criteria for ordinary self-dual and Hermitian self-dual λ-constacyclic codes as special cases.The Hermitian specialization assumes e is even.Source: §6 · p. 18

7 Examples

• In the repeated-root θ²-constacyclic example, a Hermitian self-dual code exists although no self-dual code exists; the λ = 1 cyclic example is both.For the θ²-constacyclic code, the q-coset function value is ϕ(1) = 1; for the cyclic example, ψ(1) = 1.Source: §7 · p. 19
• The θ12-constacyclic code is neither self-dual nor Hermitian self-dual, but is isometrically 3h-self-dual for every h from 0 through 4.It is also isometrically Hermitian self-dual.Source: §7 · pp. 19–20
• A semisimple negacyclic code over F9 has parameters [4, 2, 3] and is both self-dual and Hermitian self-dual.The example uses length 4 over F9.Source: §7 · p. 20
• The examples note that self-dual and Hermitian self-dual constacyclic codes may both exist even when no constacyclic code is both.Source: §7 · p. 20
• The semisimple negacyclic codes Cϕ−1 and Cϕ−5 over F25 are respectively 50-self-dual but not 51-self-dual, and 51-self-dual but not 50-self-dual.Here 50-self-dual means self-dual, while 51-self-dual means Hermitian self-dual.Source: §7 · p. 21