Self-conjugate-reciprocal irreducible monic factors of x − 1 over finite fields and their applications

Self-reciprocal and self-conjugate-reciprocal irreducible factors of x − λ and their applications

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In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugatereciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of x n − λ, λ ∈ F * q , over F q , which are just open questions posed by Boripan et al (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.2010 Mathematics Subject Classification.

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“…Most recently, Boripan et al [1] introduced the concept of self-conjugate-reciprocal (SCR) polynomials over F q 2 . They gave a characterization of SCR irreducible factors of x n − 1 and presented a recursive formula for the number of such factors.…”

Möbius–frobenius Maps on Irreducible Polynomials

“…Then σ i naturally extends to the polynomial ring F q n [x] and, for simplicity, we denote this extension by σ i . In particular, [2] Möbius-Frobenius maps 67 the projective semilinear group PΓL(2, q n ) = PGL(2, q n ) ⋊ Gal(F q n /F q ) induces maps of Möbius-Frobenius type on the set of monic irreducible polynomials over F q n . Namely, for f ∈ I k := I(q n , k) with k ≥ 2 and g ∈ PΓL(2, q n ) with g = [A, σ i ], we define the composition [A, σ i ] * f = [A] • (σ i ( f )).…”

Self-dual and complementary dual abelian codes over Galois rings

“…Over some finite rings, a characterization of self-dual cyclic, constacyclic and abelian codes has been done (see, for example, [1], [7], [16], [17], [24], and [26]). In [1], [5], [4] and [23], characterization and enumeration of Euclidean and Hermitian self-dual cyclic codes over finite chain rings have been discussed. Euclidean complementary dual cyclic codes over finite fields have been studied in [27].…”

“…Euclidean complementary dual cyclic codes over finite fields have been studied in [26]. Recently, they have been generalized to Euclidean and Hermitian complementary dual abelian codes over finite fields in [5]. The complete characterization and enumeration of complementary dual abelian codes over finite fields have been established in the said paper.In this paper, we focus on abelian codes over Galois rings GR(p r , s), i.e., ideals in the group ring GR(p r , s)[G] of an abelian group G over a Galios ring GR(p r , s).…”