A circulant approach to skew-constacyclic codes

Abstract

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x; θ] modulo the left ideal generated by a polynomial of the type x n − a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This recovers the well-known result that the dual of a skew-constacyclic code is a constacyclic code again. Special attention is paid to the case where x n − a is central.

Finite nonassociative algebras obtained from skew polynomials and possible applications to (<i>f</i>, <i>σ</i>, <i>δ</i>)-codes

“…. , t m−1 is the circulant matrix defined in [16,Definition 3.1], see also Section 4. This is already part of [8,Theorem 2] and generalizes [13,Proposition 1]: it shows that sometimes h is a parity check polynomial for C also when f is not two-sided.…”

“…, t m−1 . Therefore [16,Theorem 3.6] states that for associative algebras S f , right multiplication gives the right regular representation of the algebra, so that the product of the matrix representing R h , and the one representing R g , for any 0 = h ∈ S f , 0 = g ∈ S f , is the matrix representing R hg in S f . The fact that γ is injective and additive is observed in [16, Moreover, the matrix equation in [16, Theorem 5.6 (1)] can be read as follows: if t n − a = hg and c = γ(a, g), then the matrix representing the right multiplication with the element g(t) ∈ R n in the algebra S f where f (t) = t n − a ∈ F q [t; σ], equals the transpose of the matrix representing the right multiplication with an element g ♯ (t) ∈ S f1 where f 1 (t) = t n − c −1 ∈ F q [t; σ].…”

“…We have g ∈ ker(L h ′ ) = {u ∈ R | deg(u) < m and hu ∈ Rf } and h ∈ ker(L g ) = {u ∈ R | deg(u) < m and gu ∈ Rf }. If f is two-sided, ker(L h ′ ) = gS f and ker(L g ) = h ′ S f .Furthermore, (iii) and (iv) tie in with or generalize (4),(5) in[16, Theorem 6.6].…”

Division algebras and MRD codes from skew polynomials

“…Skew polynomials have been successfully used in constructions of division algebras (mostly semifields) and linear codes [2,3,4,11,21,22,23], in particular building space-time block codes (STBCs) [24] and maximum rank distance (MRD) codes [27,28].…”

How to obtain lattices from $$(f,\sigma ,\delta )$$ ( f , σ , δ ) -codes via a generalization of Construction A

“…As recently several classes of cyclic ( f, σ, δ)-codes were constructed with a better minimal distance for certain lengths than previously known codes (e.g., see [5][6][7][8][9][10][11]15,20,27,36,59]), ( f, σ, δ)-codes become increasingly important. These codes employ skew polynomial rings S[t; σ, δ] where S is a unital ring, σ an injective endomorphism of S and δ a left σ -derivation of S, and are built by choosing a monic polynomial f ∈ S[t; σ, δ] of degree m, and some monic right divisor g of f [13].…”