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Abstract: Abstract In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups \(M_{11}\) , \(M_{12}\) , \(M_{22}\) , \(M_{23}\) and \(M_{24}\) . From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to \(M_{22}\) , whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the codes. PubDate: 2024-03-01

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Abstract: Abstract Let \(\mathbb {F}_q\) be the finite field of q elements, and \(\mathbb {F}_{q^n}\) its extension of degree n. A normal basis of \(\mathbb {F}_{q^n}\) over \(\mathbb {F}_q\) is a basis of the form \(\{\sigma ^i(\alpha ):i=0,\dots ,n-1\}\) where \(\sigma \) denotes the Frobenius automorphism of \(\mathbb {F}_{q^n}\) over \(\mathbb {F}_q\) and \(\alpha \in \mathbb {F}_{q^n}\) . Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. Some problems on normal bases are characterized in terms of linearized polynomials. We show that such problems can be finally reduced to the determination of the irreducible factors of \(x^n-1\) over \(\mathbb {F}_q\) . Using this approach, we extend the known results with alternative as well as relatively elementary proofs. And we believe that this approach can be used to deal with other similar problems. PubDate: 2024-03-01

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Abstract: Abstract We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order 1. We present a new symbolic procedure solving this problem with a computable complexity based on the computation of suitable Sturm sequences. For a general Pfaffian function, we assume the existence of an oracle to determine the sign that a function of the class takes at a real algebraic number. As a by-product, we obtain a new oracle-free effective algorithm solving the same problem for univariate E-polynomials based on techniques that are simpler than the previous ones, and we apply it to solve a similar decision problem in the multivariate setting. Finally, we introduce a notion of Thom encoding for zeros of an E-polynomial and describe an algorithm for their computation. PubDate: 2024-03-01

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Abstract: Abstract In this paper we present the Multidimensional Byzantine Agreement (MBA) Protocol, a leaderless Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant information regarding a set of observed events. The consensus process is carried out in parallel on each component, and the output is a vector whose components are either values with wide agreement in the network (even if no individual node agrees on every value) or a special value \(\bot \) that signals irreconcilable disagreement. The MBA Protocol is probabilistic and its execution halts with probability 1, and the number of steps necessary to halt follows a Bernoulli-like distribution. The design combines a Multidimensional Graded Consensus and a Multidimensional Binary Byzantine Agreement, the generalization to the multidimensional case of two protocols presented by Micali et al. (SIAM J Comput 26(4):873–933, 1997; Byzantine agreement, made trivial, 2016). We prove the correctness and security of the protocol assuming a synchronous network where less than a third of the nodes are malicious. PubDate: 2024-03-01

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Abstract: Abstract In this paper, we introduce \({\mathbb {F}}_{q^{2}}\) -double cyclic codes of length \(n=r+s,\) where \({\mathbb {F}}_{q^{2}}\) is the Galois field of \(q^{2}\) elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any \({{\mathbb {F}}_{q^{2}} }\) -double cyclic code. For any \({\mathbb {F}}_{q^{2}}\) -double cyclic code \({\mathcal {C}},\) we will define the Euclidean dual code \({\mathcal {C}}^{\perp }\) based on the Euclidean inner product and the Hermitian dual code \({\mathcal {C}} ^{\perp _{H}}\) based on the Hermitian inner product. We will construct a relationship between \({\mathcal {C}}^{\perp }\) and \({\mathcal {C}}^{\perp _{H}}\) and then find the generator polynomials for the Hermitian dual code \({\mathcal {C}}^{\perp _{H}}.\) As an application of our work, we will present examples of optimal parameter linear codes over the finite field \({\mathbb {F}} _{4}\) and also examples of optimal quantum codes that were derived from \({\mathbb {F}}_{4}\) -double cyclic codes using the Hermitian inner product. PubDate: 2024-03-01

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Abstract: Abstract The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with \(d^{1.5 + o (1)}\) for input polynomials of degree d, and with the square of the bit size of the ground field. It relies on a variant of the Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth. PubDate: 2024-03-01

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Abstract: Abstract Kostant’s weight q-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. The q-analog of the partition function is a polynomial-valued function defined by \(\wp _q(\xi )=\sum _{i=0}^k c_i q^i\) , where \(c_i\) is the number of ways the weight \(\xi\) can be written as a sum of exactly i positive roots of a Lie algebra \({\mathfrak {g}}\) . The evaluation of the q-multiplicity formula at \(q = 1\) recovers the multiplicity of a weight in an irreducible highest weight representation of \({\mathfrak {g}}\) . In this paper, we specialize to the Lie algebra \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\) and we provide a closed formula for the q-analog of Kostant’s partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\) ), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of \({{\mathfrak {s}}}{{\mathfrak {p}}}_6({\mathbb {C}})\) . Throughout this work, we provide code to facilitate these computations. PubDate: 2024-03-01

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Abstract: Abstract Let a and m be positive integers and \(\lambda\) be any unit in \({{\,\mathrm{GR}\,}}(p^a,m)\) of the form \(\lambda =(\sigma _0+p\sigma _1+p^2z)\) , where \(\sigma _0,\sigma _1 \in \mathcal {T}(p,m)\smallsetminus \{0\}\) and \(z \in {{\,\mathrm{GR}\,}}(p^a,m)\) . The symbol-pair distance of all such \(\lambda\) -constacyclic codes over \({{\,\mathrm{GR}\,}}(p^a,m)\) of length \(p^s\) are determined. As an application, we identify all maximum distance separable (MDS) \(\lambda\) -constacyclic codes of length \(p^s\) over \({{\,\mathrm{GR}\,}}(p^a,m)\) with respect to the symbol-pair distance. We give numerous examples of newly constructed MDS symbol-pair codes, i.e., new optimal symbol-pair codes with respect to the Singleton bound. PubDate: 2024-03-01

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Abstract: Abstract This paper extends the concept of weighted point clouds and weighted simplicial complexes by introducing product point clouds and product simplicial complexes within a commutative ring with unity. Within an integral domain, the introduction of a weighted product chain group, along with the induced product weighted homomorphism and weighted product boundary maps, leads to significant outcomes and findings. To explore the algebraic characteristics of a weighted product structure, we introduce the concept of weighted product homology. This homology considers the relationship of weights assigned to elements within the structure and their impact on the structure’s underlying algebraic properties. PubDate: 2024-02-09

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Abstract: Abstract The weight hierarchy of a linear code have been an important research topic in coding theory since Wei’s original work in 1991. In this paper, choosing \(D=\Big \{(x,y)\in \Big ({\mathbb {F}}_{p^{s_1}}\times {\mathbb {F}}_{p^{s_2}}\Big )\Big \backslash \{(0,0)\}: f(x)+\text {Tr}_1^{s_2}(\alpha y)=0\Big \}\) as a defining set, where \(\alpha \in {\mathbb {F}}_{p^{s_2}}^*\) , f(x) is a quadratic form over \({\mathbb {F}}_{p^{s_1}}\) with values in \({\mathbb {F}}_p\) and f(x) can be non-degenerate or not, we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies completely. Most of the codes can be used in secret sharing schemes. PubDate: 2024-02-04

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Abstract: Abstract In this paper, a necessary condition which is sufficient as well for a pair of constacyclic 2-D codes over a finite commutative ring R to be an LCP of codes has been obtained. Also, a characterization of non-trivial LCP of constacyclic 2-D codes over R has been given and total number of such codes has been determined. The above results on constacyclic 2-D codes have been extended to constacyclic 3-D codes over R. The obtained results readily extend to constacyclic n-D codes, \(n \ge 3\) , over finite commutative rings. Furthermore, some results on existence of non-trivial LCP of constacyclic 2-D codes over a finite chain ring have been obtained in terms of its residue field. PubDate: 2024-01-19

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Abstract: Abstract An almost perfect non-linear (APN) function over \(\mathbb {F}_{2^n}\) is called exceptional APN if it remains APN over infinitely many extensions of \(\mathbb {F}_{2^n}\) . Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions. PubDate: 2024-01-18

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Abstract: Abstract Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension. PubDate: 2024-01-07

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Abstract: Abstract A relationship between a parametric ideal and its generic Gröbner basis is discussed, and a new stability condition of a Gröbner basis is given. It is shown that the ideal quotient of the parametric ideal by the ideal generated using the generic Gröbner basis provides the stability condition. Further, the stability condition is utilized to compute a comprehensive Gröbner system, and hence as an application, we obtain a new algorithm for computing comprehensive Gröbner systems. The proposed algorithm has been implemented in the computer algebra system Risa/Asir and experimented with a number of examples. Its performance (execution timings) has been compared with the Kapur-Sun-Wang’s algorithm for computing comprehensive Gröbner systems. PubDate: 2024-01-01

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Abstract: Abstract This note is just a modest contribution to prove several classical results in Combinatorics from notions of Duality in some Artinian K-algebras (mainly through the Trace Formula), where K is a perfect field of characteristics not equal to 2. We prove how several classic combinatorial results are particular instances of a Trace (Inversion) Formula in finite \(\mathbb {Q}\) -algebras. This is the case with the Exclusion-Inclusion Principle (in its general form, both with direct and reverse order associated to subsets inclusion). This approach also allows us to exhibit a basis of the space of null t-designs, which differs from the one described in Theorem 4 of Deza and Frankl (Combinatorica 2:341–345, 1982). Provoked by the elegant proof (which uses no induction) in Frankl and Pach (Eur J Comb 4:21–23, 1983) of the Sauer–Shelah–Perles Lemma, we produce a new one based only in duality in the \(\mathbb {Q}\) -algebra \(\mathbb {Q}[V_n]\) of polynomials functions defined on the zero-dimensional algebraic variety of subsets of the set \([n]:=\{1,2,\ldots , n\}\) . All results are equally true if we replace \(\mathbb {Q}[V_n]\) by \(K[V_n]\) , where K is any perfect field of characteristics \(\not =2\) . The article connects results from two fields of mathematical knowledge that are not usually connected, at least not in this form. Thus, we decided to write the manuscript in a self-contained survey-like style, although it is not a survey paper at all. Readers familiar with Commutative Algebra probably know most of the proofs of the statements described in section 2. We decided to include these proofs for those potential readers not so familiar with this framework. PubDate: 2024-01-01

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Abstract: Abstract In this paper we introduce and analyze a discrete SIS epidemic model for a homogeneous population. As a discretization method the strictly positive scheme was chosen. The presented model is built from its continuous counterpart known from literature. We firstly present basic properties of the system. Later we discuss local stability of stationary states and global stability for the disease-free stationary state. The results for this state are expressed with the use of the basic reproduction number. The main conclusion from our work is that conditions for stability of the stationary states do not depend on the step size of the discretization method. This fact stays in contrary to other discrete models analyzed in our previous papers. Theoretical results are accomplished with numerical simulations. PubDate: 2024-01-01

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Abstract: Abstract The label code of a lattice plays a key role in the characterization of the lattice. Every lattice \(\Lambda\) can be described in terms of a label code L and an orthogonal sublattice \(\Lambda '\) such that \(\Lambda /\Lambda '\cong L\) . We identify the binomial ideal associated to an integer lattice and then establish a relation between the ideal quotient of the lattice and its label code. Furthermore, we present the Gröbner basis of the well-known root lattice \(D_n\) . As an application of the relation \(I_{\Lambda }=I_{\Lambda '}+I_{L}\) , where \(I_{\Lambda },I_{\Lambda '}\) and \(I_L\) denote binomial ideals associated to \(\Lambda ,~\Lambda '\) and L, respectively, a linear label code of \(D_n\) is obtained using its Gröbner basis. PubDate: 2024-01-01

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Abstract: Abstract We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations \(\sigma\) . Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of \(\sigma\) are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial. PubDate: 2023-10-28 DOI: 10.1007/s00200-023-00628-0