The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly

Let \(\gamma = 4z-1\) be an unit of Type \((*^{-})\) of the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\). The \(\gamma\)-constacyclic codes of length \(2^s\) over the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\) are precisely the ideals \(\langle (x +1)^i \rangle\), \(0 \le i \le 2^sa\) of the chain ring \(\mathfrak {R}(a,m, \gamma ) = \dfrac{{{\,\mathrm{GR}\,}}(2^a,m)[x]}{\langle {x^{2^s}} - \gamma

Let \(m=8k\) and \(\alpha\) be a primitive element of the finite field \({{\mathbb {G}}{\mathbb {F}}}(2^m)\), where \(k\ge 2\) is an integer. In this paper, a class of binary cyclic codes \({{\mathcal {C}}}_{(u,v)}\) of length \(2^m-1\) with two nonzeros \(\alpha ^{-u}\) and \(\alpha ^{-v}\) is studied, where \((u,v)=(1,(2^{m}-1)/17)\). It turns out that \({{\mathcal {C}}}_{(u,v)}\) has parameters

In this paper, we give the exact number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes of length \(n=r+s,\) for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable \({{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n and then a formula for the number of non-separable \({{\mathbb {Z}} _{2}{{\mathbb

In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if \({\text {Tor}}_{1}^{R}(A, R/I)= 0\) for any maximal left ideal I of R. A right module B is said to be max-cotorsion if \({\text {Ext}}^{1}_{R}(A,B)=0\) for any max-flat right module A. We characterize some

In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti–Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized Hamming weights of such codes.

Let R be a ring such that \(R=R_1+R_2\), where \(R_1\) is a PI subring of R and \(R_2\) is an additive subgroup of R which satisfies a polynomial identity. We prove that if for some integer \(n\ge 1\) either \((R_1R_2)^n \subseteq R_1\) or \((R_2R_1)^n \subseteq R_1\), then R is a PI ring.

For \(m\ge 4\) even, the duals of p-ary codes, for any prime p, from adjacency matrices for the m-ary 2-cube \(Q^m_2\) are shown to have subcodes with parameters \([m^2,2m-2,m]\) for which minimal PD-sets of size \(\frac{m}{2}\) are constructed, hence attaining the full error-correction capabilities of the code, and, as such, the most efficient sets for full permutation decoding.

The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions

In this paper, we prove that flag-transitive automorphism groups of 2-designs with \(10^3\ge \lambda \ge (r,\lambda )^2\) are point-primitive of affine or almost simple type.

In this paper we derive formulas for the scalar multiplication by n map, denoted [n], on the Hessian model of elliptic curve. This enables to characterize n-torsion points on this curve. The computation involves three families of polynomials \(P_n\), \(Q_n\) and \(V_n\) and we show some properties on the coefficients and degrees of these polynomials. We also show some functional equations satisfied

Let \(\gamma\) be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that \(f(x)x^{-r}\) is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie–Hellman mapping \(d(\gamma ^a)=\gamma ^{a^2}\), \(a=0,1,\ldots ,n-1\), and show that any mapping of small index coincides

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group \({\mathbb {Z}}_{p^e}\). They exist for all lengths

Let \({\mathbb {F}}_q\) be a finite field with q elements and G be a finite abelian group. In this work we gave conditions to ensure that a code in \({\mathbb {F}}_qG\) is a one-weight code in the case when G is a cyclic group with n elements, such that \({\text {gcd}}(n,q) = 1\), and also when G is an abelian group.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted

We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most previously-known greedy algorithms that generate binary de Bruijn sequences are particular cases of our algorithm.

Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The codes constructed are (3, 3)-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the codes obtained and present

The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite

Traditional global stability measure for sequences is hard to determine because of large search space. We propose the k-error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the k-error linear complexity is identical to the k-error linear complexity within a zone, while the length of a zone is much smaller

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler’s algorithm, is designed for them.

We construct an infinite family of commutative rings \({R_{q,\varDelta }}\) and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from \({R_{q,\varDelta }}\) to vectors over the residue finite field of q elements and use it to relate codes over \({R_{q,\varDelta }}\) to codes over the finite field \({\mathbb {F}}_q\). Finally, we determine the parameters

Quasi-symmetric (36, 16, 12) designs with intersection numbers \(x=6\), \(y=8\) that cannot be embedded in symmetric (64, 28, 12) designs as residuals are constructed.

The support poset of a monomial ideal \(I\subseteq {{\mathbf {k}}}[x_1,\ldots ,x_n]\) encodes the relation between the variables \(x_1,\ldots ,x_n\) and the minimal monomial generators of I. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets P for which we can explicitly find at least one monomial ideal \(I_P\) such that P is the support

InfoMod is a software devoted to the modular group, \(\mathrm {PSL}_2 (\mathbf {Z})\). It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss’ indefinite binary quadratic forms and their classes in terms

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth

The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at \(q=1\), gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra \(\mathfrak {sl}_4(\mathbb {C})\)

For a prime p and positive integers m, n, let \({{\mathbb {F}}}_q\) be a finite field with \(q=p^m\) elements and \({{\mathbb {F}}}_{q^n}\) be an extension of \({{\mathbb {F}}}_q.\) Let h(x) be a polynomial over \({{\mathbb {F}}}_{q^n}\) satisfying the following conditions: (i) \({\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x)\); (ii) For any \(s \in {{\mathbb {F}}}_{q}\), h(x)

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over \(\mathbb {C}\), this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of

In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases.

Symplectic self-orthogonal codes over finite fields are an important class of linear codes in coding theory, which can be used to construct quantum codes. In this paper, characterizations of symplectic self-orthogonal codes over finite fields \(F_{q}\) are given. A necessary and sufficient condition for determining symplectic self-orthogonal codes is obtained. Several classes of symplectic self-orthogonal

Let q be a prime power.

An associative memory is a special type of artificial neural network that has the purpose of store input patterns with their corresponding output patterns and efficiently recall a pattern from a noise-distorted version. Presented in this article is a new framework for constructing a binary associative memory model based on two new autoinverse operations called extended XOR/XNOR; these new operations

We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it

Rigid graph theory is an active area with many open problems, especially regarding embeddings in \({\mathbb R}^d\) or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system’s complex solutions

The F4 algorithm re-imagines Buchberger’s algorithm as the row reduction of a Macaulay matrix: each row corresponds to a polynomial; each reduction of one row by another corresponds to one step of reducing an S-polynomial; and any row that completes reduction with a new pivot position corresponds to a new element of the basis. On the other hand, each column corresponds to a term, so that while it is

When a pairing \(e: {\mathbb {G}}_1 \times {\mathbb {G}}_2 \rightarrow {\mathbb {G}}_{T}\), on an elliptic curve E defined over a finite field \({\mathbb {F}}_q\), is exploited for an identity-based protocol, there is often the need to hash binary strings into \({\mathbb {G}}_1\) and \({\mathbb {G}}_2\). Traditionally, if E admits a twist \({\tilde{E}}\) of order d, then \({\mathbb {G}}_1=E({\mathbb

Dynamic Gröbner Basis algorithms aim to obtain small reduced Gröbner Bases in terms of number of polynomials and monomials. Dynamic algorithms allowing previous leading monomials to change during the execution, called unrestricted algorithms, are underexplored in the literature and the only previous unrestricted algorithm was not implemented. In this paper, we introduce a definition of neighborhoods

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form \({\mathbf{x}}\cdot \pi ({\mathbf{y}})\oplus h({\mathbf{y}})\) [Maiorana–McFarland

In this paper, we investigate all coset leaders of primitive BCH codes for \(\delta\) in the range \(1\le \delta \le q^\frac{m+7}{2}\), which extends Liu and Shi’s results. Besides, we also generalize Shi’s results by proposing the maximum designed distance of non-narrow-sense(\(b=k_2q^2+k_1q+k_0\)) primitive BCH codes which can contain their Euclidean dual. At the end, we calculate the dimension of

Interplay between coding theory and combinatorial t-designs has attracted a lot of attention. It is well known that the supports of all codewords of a fixed Hamming weight in a linear code may hold a t-design. In this paper, we first settle the weight distributions of two classes of linear codes, and then determine the parameters of infinite families of 2-designs held in these codes.

In this paper we study a family of Legendre sequences and its pseudo-randomness in terms of their family complexity. We present an improved lower bound on the family complexity of a family based on the Legendre symbol of polynomials over a finite field. The new bound depends on the Lambert W function and the number of elements in a finite field belonging to its proper subfield. Moreover, we present

In this paper we consider in detail the composition of an irreducible polynomial with \(X^2\) and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an irreducible polynomial of degree n and order \(2^rt\) with t odd, the construction produces \(ord_t(2)\) irreducible polynomials of degree n and order t. The construction

Lechuga and Murillo showed that a non-oriented, simple, connected, finite graph G is k-colourable if and only if a certain pure Sullivan algebra associated to G and k is not elliptic. In this paper, we extend this result to simplicial complexes by means of several notions of colourings of these objects.

Let s be a prime power and \( {{{\mathbb {F}}}}_q\) be a finite field with s elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form $$\begin{aligned} b(x^q+ax+\delta )^{1+\frac{i(q^2-1)}{d}}+c(x^q+ax+\delta )^{1+\frac{j(q^2-1)}{d}}+L(x) \end{aligned}$$ over \( {{{\mathbb {F}}}}_{q^2}\) with \(a^{1+q}=1, q\equiv \pm 1\pmod {d}\) and

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