Error-correcting codes that can effectively encode and decode messages of distinct lengths while maintaining a constant blocklength are considered. It is known conventionally that a k-dimensional block code of length n defined over \(\texttt {GF}(q^{n})\) is designed to encode a k-symbol user data in to an n-length codeword, resulting in a fixed-rate coding. In contrast, considering \(q=p^{\lambda

In this paper, we introduce double Quadratic Residue Codes (QRC) of length \(n=p+q\) for prime numbers p and q in the ambient space \({{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.\) We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when \(p=q.\) We give the main properties for

In this paper, we provide new methods and algorithms to construct Euclidean self-dual codes over large finite fields. With the existence of a dual basis, we study dual preserving linear maps, and as an application, we use them to construct self-orthogonal codes over small finite prime fields using the method of concatenation. Many new optimal self-orthogonal and self-dual codes are obtained.

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this

We study the equivalence between the ring learning with errors and polynomial learning with errors problems for cyclotomic number fields, namely: we prove that both problems are equivalent via a polynomial noise increase as long as the number of distinct primes dividing the conductor is kept constant. We refine our bound in the case where the conductor is divisible by at most three primes and we give

For any odd positive integer n, we express cyclic codes over \({\mathbb {Z}}_4\) of length 4n in a new way. Based on the expression of each cyclic code \({\mathcal {C}}\), we provide an efficient encoder and determine the type of \({\mathcal {C}}\). In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over \({\mathbb {Z}}_4\) of length 4n and correct

Let p be an odd prime, and \(\delta\) be an arbitrary unit of the finite chain ring \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m} \,\, (u^2=0)\). The Hamming distances of all \(\delta\)-constacyclic codes of length \(2p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) are completely determined. We provide some examples from which some codes have better parameters than the existing ones. As applications

In 1990 Cooper suggested to use Groebner bases’ computations to decode cyclic codes and his idea gave rise to many research papers. In particular, as proved by Sala-Orsini, once defined the polynomial ring whose variables are the syndromes, the locations and the error values and considered the syndrome ideal, only one polynomial of a lexicographical Groebner basis for such ideal is necessary to decode

Let f(u) and g(v) be two polynomials, not both linear, which split into distinct linear factors over \({\mathbb {F}}_{q}\). Let \({\mathcal {R}}={\mathbb {F}}_{q}[u,v]/\langle f(u),g(v),uv-vu\rangle \) be a finite commutative non-chain ring. In this paper, we study general skew cyclic codes and \(\theta _t\)-skew constacyclic codes over the ring \({\mathcal {R}}\) where \(\theta _t\) is an automorphism

In this paper, we present some work towards a complete characterization of Hilbert quasi-polynomials of graded polynomial rings. In this setting, a Hilbert quasi-polynomial splits in a polynomial F and a lower degree quasi-polynomial G. We completely describe the periodic structure of G. Moreover, we give an explicit formula for the \((n-1)\)th and \((n-2)\)th coefficient of F, where n denotes the

In 1994, Moss Sweedler’s dog proposed a cryptosystem, known as Barkee’s Cryptosystem, and the related cryptanalysis. Its explicit aim was to dispel the proposal of using the urban legend that “Gröbner bases are hard to compute”, in order to devise a public key cryptography scheme. Therefore he claimed that “no scheme using Gröbner bases will ever work”. Later, further variations of Barkee’s Cryptosystem

Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability

In this paper, we first introduce a trigonometric approach for Dickson polynomials of the first and the second kinds over fields of characteristic two. Employing the proposed concepts, we revisit known properties of such polynomials. Additionally, we derive new results regarding the fixed points and the involutive behavior of Dickson polynomials of the second kind over \({\mathbb {F}}_{2^n}\).

RS-like locally recoverable (LRC) codes have construction based on the classical construction of Reed–Solomon (RS) codes, where codewords are obtained as evaluations of suitably chosen polynomials. These codes were introduced by Tamo and Barg (IEEE Trans Inf Theory 60(8):4661–4676, 2014) where they assumed that the length n of the code is divisible by \(r+1\), where r is the locality of the code. They

In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block \(a_0a_1\ldots a_{n-1}\) is extended into a block \(a_0a_1\ldots a_{n-1}d_0d_1\ldots d_{n-1}\), where the redundant characters \(d_0, d_1, \ldots , d_{n-1}\) are defined with \(d_i=a_i*a_{i+1}*a_{i+2}\), where \(*\) is a linear quasigroup operation and the operations in the indexes are modulo n.

This paper is a continuation of ideas presented by Bagirmaz (Appl Algebra Eng Commun Comput 30(4):285–297, 2019). Indeed, we introduce the notion of near approximations in a ring, which is an extended notion of a rough approximations in a ring. Then we define lower and upper near subrings based on ideals in a ring and give some properties of such subrings. Furthermore, we obtain a comparison between

Recently, Kim et al. proposed a modified Dual-Ouroboros public-key encryption (\({\textsf{PKE}}\)) using Gabidulin codes to overcome the limitation of having decryption failure in the original Dual-Ouroboros using low rank parity check codes. This modified Dual-Ouroboros \({\textsf{PKE}}\) using Gabidulin codes is proved to be IND–CPA secure, with very compact public key size of 738 bytes achieving

A numerical semigroup S is closed under addition of its divisors (\({\mathcal {C}}\)-semigroup) if the following condition holds: if \(s\in S\setminus \{0\}\) and d is a non trivial divisor of s then \(s+d\in S\). In this paper we prove that the set of all \({\mathcal {C}}\)-semigroups is a Frobenius variety. As a consequence, we give algorithms to compute the set \({\mathcal {C}}\)-semigroups with

We apply fundamental ideas from algebraic topology in mathematics to the digital world in computer science. We develop digital H-spaces and digital H-functions between the digital H-spaces with digital multiplications, and construct a necessary and sufficient condition for a digital H-space (or a digital homotopy associative H-space, a digital homotopy commutative H-space, or a digital H-group) to

In this work, we study linear codes over the ring \(\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)\) and their weight enumerators, where \(v^2=v\). We first give the structure of the ring and investigate linear codes over this ring. We also define two weights called Lee weight and Gray weight for these codes. Then we introduce two Gray maps from \(\mathbb {F}_2 \times (\mathbb {F}_2+v\mathbb {F}_2)\)

We consider \(m \times s\) matrices (with \(m\ge s\)) in a real affine subspace of dimension n. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently

Let \({M}_{m,n}\) denote the space of \(m\times n\) matrices with entries in the formal power series ring \(K[[x_1,\ldots , x_s]], K\) an arbitrary field. We consider different groups G acting on \(M_{m,n}\) by formal change of coordinates, combined with the multiplication by invertible matrices. This includes right and contact equivalence of functions, mappings, and ideals. A matrix A is called finitely