Type II $${\mathbb {Z}}_4$$ Z 4 -codes are a class of self-dual $${\mathbb {Z}}_4$$ Z 4 -codes with Euclidean weights divisible by eight. A Type II $${\mathbb {Z}}_4$$ Z 4 -code of length n is extremal if its minimum Euclidean weight is equal to …

Given an algebraic germ of a plane curve at the origin, in terms of a bivariate polynomial, we analyze the complexity of computing an irreducible decomposition up to any given truncation order. With a suitable representation of the irreducible …

In this paper, two classes of permutation pentanomials over finite fields $$\mathbb {F}_{p^{2m}}$$ F p 2 m are investigated by transforming the permutation property of polynomials to verifying that some low-degree equations has no solutions in the …

In this paper, we define the complex-type k-Padovan numbers and then give the relationships between the $$\left( 1,k-1\right)$$ 1 , k - 1 -bonacci numbers, the k -Padovan numbers and the complex-type k-Padovan numbers by matrix method. In …

Let $$\mathbb{F}_q$$ F q be a finite field of characteristic p. In this paper we prove that the c-Boomerang Uniformity, $$c \ne 0$$ c ≠ 0 , for all permutation monomials $$x^d$$ x d , where $$d > 1$$ d > 1 and $$p \not \mid d$$ p ∤ d , is bounded …

In order to get a better code rate, this study focuses on the construction of double skew cyclic codes over the ring $$\textrm{R}= \mathbb {F}_q+v\mathbb {F}_q$$ R = F q + v F q with $$v^2=v$$ v 2 = v , where q is a prime power. We investigate the …

The main results of this paper are in two directions. First, the family of finite local rings of length 4 whose annihilator of their maximal ideals have length 2 is determined. Second, the structure of constacyclic, reversible and DNA codes over …

Let $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ …

Lattice theory has shown to be useful in information theory, and rotated lattices with high modulation diversity have been extensively studied as an alternative approach for transmission over a Rayleigh fading channel, where the performance …

In this article, we deal with additive codes over the Frobenius ring $${\mathcal {R}}_{2}{\mathcal {R}}_{3}:=\frac{{\mathbb {Z}}_{2}[u]}{\langle u^2 \rangle }\times \frac{{\mathbb {Z}}_{2}[u]}{\langle u^3 \rangle }$$ R 2 R 3 : = Z 2 [ u ] ⟨ u 2 ⟩ …

We describe the notion of stability of coherent systems as a framework to deal with redundancy. We define stable coherent systems and show how this notion can help the design of reliable systems. We demonstrate that the reliability of stable …

In this paper, we give a construction of doubly even self-orthogonal codes from quasi-symmetric designs. Especially, we study orbit matrices of quasi-symmetric designs and give a construction of doubly even self-orthogonal codes from orbit …

We study additive codes with 1-rank hulls and examine their properties for various dualities of the finite field of order 4. We give several constructions of additive and linear codes with 1-rank hulls. We also relate these codes to additive …

Let $${{\mathbb {K}}}$$ K be any field, let $$X\subset {\mathbb P}^{k-1}$$ X ⊂ P k - 1 be a set of $$n$$ n distinct $${{\mathbb {K}}}$$ K -rational points, and let $$a\ge 1$$ a ≥ 1 be an integer. In this paper we find lower bounds for the minimum …

Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let p be an odd prime and let s and m be positive integers. In this paper, we …

Permutation polynomials with sparse forms over finite fields attract researchers’ great interest and have important applications in many areas of mathematics and engineering. In this paper, by investigating the exponents (s, i) and the …

Let $$k>1$$ k > 1 be a fixed integer. In Gupta and Ray (Cryptography and Communications 7: 257–287, 2015), proved the non existence of $$2^k \times 2^k$$ 2 k × 2 k orthogonal circulant MDS matrices and involutory circulant MDS matrices over finite …

Consider a multivariate polynomial $$f \in K [x_1, \ldots , x_n]$$ f ∈ K [ x 1 , … , x n ] over a field K, which is given through a black box capable of evaluating f at points in $$K^n$$ K n , or possibly at points in $$A^n$$ A n for any K-algebra …

Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptosystems. A …

Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilator-ideal graph of R is defined as the graph $$\mathrm{A_I}(R)$$ A I ( R ) with the vertex set $$A(R)^*=A(R)\setminus \{0\}$$ A ( R ) ∗ …