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Abstract: Abstract 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. PubDate: 2022-06-01
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Abstract: Abstract 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 the Euclidean dual containing non-narrow-sense primitive BCH codes and construct some quantum BCH codes. PubDate: 2022-06-01
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Abstract: Abstract 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 codes are constructed. Furthermore, the symplectic weight distributions of some new classes of binary symplectic self-orthogonal codes are completely determined. PubDate: 2022-06-01
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Abstract: Abstract 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 are generated from the XOR/XNOR operations, respectively. Two types of associative memory are generated with this model: the max type (XOR-AM max), which is constructed with the maximum of the extended XOR operation, and the min type (XOR-AM min), which is constructed with the minimum of the extended XNOR operation. The XOR-AM max exhibits tolerance against the presence of patterns distorted by dilative noise, whereas the XOR-AM min exhibits tolerance against the presence of patterns distorted by erosive noise; both types of memory converge in a single step, use the same extended XOR/XNOR operator for learning and recalling phases, operate in heteroassociative and autoassociative modes, and show infinite storage capacity for the autoassociative mode. Finally, computer simulation results are presented for the new memories based on the extended XOR/XNOR (XOR-AM), which have better or equal performance compared to other associative memories. For the experiments with mixed noise, the conditions established by the kernel method proposed by Ritter for Morphological Associative Memories were conserved, and the solution algorithm proposed by Hattori for the construction of the kernel patterns of these memories was modified. PubDate: 2022-06-01
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Abstract: Abstract 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 \({\mathbb {G}}_1=E({\mathbb {F}}_q) \cap E[r]\) , where r is a prime integer, and \({\mathbb {G}}_2={\tilde{E}}({\mathbb {F}}_{q^{k/d}}) \cap {\tilde{E}}[r]\) , where k is the embedding degree of E w.r.t. r. The standard approach for hashing into \({\mathbb {G}}_2\) is to map to a general point \(P \in {\tilde{E}}({\mathbb {F}}_{q^{k/d}})\) and then multiply it by the cofactor \(c=\#{\tilde{E}}({\mathbb {F}}_{q^{k/d}})/r\) . Usually, the multiplication by c is computationally expensive. In order to speed up such a computation, two different methods—by Scott et al. (International conference on pairing-based cryptography. Springer, Berlin, pp 102–113, 2009) and by Fuentes-Castaneda et al. (International workshop on selected areas in cryptography)—have been proposed. In this paper we consider these two methods for BLS pairing-friendly curves having \(k \in \{12,24,30,42,48\}\) , providing efficiency comparisons. When \(k=42,48\) , the application of Fuentes et al. method requires expensive computations which were infeasible for the computational power at our disposal. For these cases, we propose hashing maps that we obtained following Fuentes et al. idea. PubDate: 2022-06-01
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Abstract: Abstract 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 ( \({\mathcal {M}}\) )] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the \({\mathcal {M}}\) class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in \(4p^k\) variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results. PubDate: 2022-06-01
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Abstract: Abstract Frequency hopping sequences (FHSs) are employed to mitigate the interferences caused by the hits of frequencies in frequency hopping spread spectrum systems. In this paper, four new constructions for FHSs are presented, including a direct construction of FHSs by using cyclotomic cosets and three extension constructions of FHSs by using units of rings and multiplicative groups of finite fields. By these constructions, a number of series of new FHSs are then produced. These FHSs are optimal with respect to the Lempel-Greenberger bound. PubDate: 2022-05-21
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Abstract: Abstract Let I be the commutative non-unital ring of order 4 defined by generators and relations. $$\begin{aligned} I=\left\langle a,b\mid 2a=2b=0,\,a^{2}=b,\,ab=0\right\rangle . \end{aligned}$$ Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths \(n=6\) , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths \(n=7\) and 8, and describe five new variants of the two building-up construction methods. We find that when \(n=8\) there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. PubDate: 2022-05-19
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Abstract: Permutation codes are studied due to their numerous applications in various applications, such as power line communications, block ciphers, and coding for storage. In this paper, we study perfect permutation codes in \(S_n\) , the set of all permutations on n elements, under the \(\ell _{\infty }\) -metric. We present some nonexistence results on perfect t-error-correcting permutation codes in \(S_n\) under the \(\ell _{\infty }\) -metric for some t and n. More precisely, we prove that there does not exist a perfect t-error-correcting code in \(S_n\) under the \(\ell _{\infty }\) -metric for t and n satisfying \(1\le t\le 3, 2t+1\le n\) or for t and n satisfying \(R_{2t+1}(n)=0,1,2t\) , where \(0\le R_d(n)< d\) is the residue when dividing the positive integer n by the positive integer d. PubDate: 2022-05-06
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Abstract: Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained by the defining sets over finite fields of odd characteristic. The parameters and weight distributions of linear codes are determined by the additive characters, multiplicative characters and Gauss sums. Further, most of linear codes obtained are minimal, which can be used to construct secret sharing schemes. PubDate: 2022-05-04
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Abstract: We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field \({\mathbb {Q}}(\zeta _n)\) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial. PubDate: 2022-04-30
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Abstract: Abstract Permutation polynomials over finite fields constitute an active research area and have important applications in many areas of science and engineering. In this paper, several classes of permutation polynomials with trace functions are presented over \(\mathbb {F}_{p^{n}} (p=2, 3)\) by investigating the number of solutions to special equations. PubDate: 2022-04-22
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Abstract: Abstract In this paper, \(\delta \) -dual codes over finite commutative semi-simple rings are defined as a generalization of dual codes over finite commutative semi-simple rings. Some properties of \(\delta \) -dual codes are given. We present necessary and sufficient conditions for a \(\lambda \) -constacyclic code of length n to be \(\delta \) -self-dual, \(\delta \) -self-orthogonal, \(\delta \) -dual-containing, \(\delta \) -LCD over finite commutative semi-simple rings. We also study the \(\delta \) -dual of skew \(\Theta \) - \(\lambda \) -constacyclic codes over finite commutative semi-simple rings. Among others, we also give necessary and sufficient conditions for a skew \(\Theta \) - \(\lambda \) -constacyclic code of any length n to be \(\delta \) -self-dual, \(\delta \) -self-orthogonal, \(\delta \) -dual-containing, \(\delta \) -LCD over finite commutative semi-simple rings. PubDate: 2022-04-20
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Abstract: Abstract Let p be an odd prime and m and s positive integers, with m even. Let further \({\mathbb {F}}_{p^m}\) be the finite field of \(p^m\) elements and \(R={\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) ( \(u^2=0\) ). Then R is a finite chain ring of \(p^{2m}\) elements, and there is a Gray map from \(R^N\) onto \({\mathbb {F}}_{p^m}^{2N}\) which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dual codes of length 2N over \({\mathbb {F}}_{p^m}\) by constructing self-dual codes of length N over R. In particular, it has been shown that one of the key problems in constructing self-dual repeated-root cyclic codes over R is to find an effective way to present precisely Hermitian self-dual cyclic codes of length \(p^s\) over R. But so far, only the number of these codes has been determined in literature. In this paper, we give an efficient way of constructing all distinct Hermitian self-dual cyclic codes of length \(p^s\) over R by using column vectors of Kronecker products of matrices with specific types. Furthermore, we provide an explicit expression to present precisely all these Hermitian self-dual cyclic codes, using binomial coefficients. PubDate: 2022-04-19
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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: 2022-03-31
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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: 2022-03-29
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Abstract: Abstract Jacobi’s results on the computation of the order and of the normal forms of a differential system are translated in the formalism of differential algebra. In the quasi-regular case, we give complete proofs according to Jacobi’s arguments. The main result is Jacobi’s bound, still conjectural in the general case: the order of a differential system \(P_{1}, \ldots , P_{n}\) is not greater than the maximum \({{\mathcal {O}}}\) of the sums \(\sum _{i=1}^{n} a_{i,\sigma (i)}\) , for all permutations \(\sigma \) of the indices, where \(a_{i,j}:=\mathrm{ord}_{x_{j}}P_{i}\) , viz. the tropical determinant of the matrix \((a_{i,j})\) . The order is precisely equal to \({{\mathcal {O}}}\) iff Jacobi’s truncated determinant does not vanish. Jacobi also gave a polynomial time algorithm to compute \({{\mathcal {O}}}\) , similar to Kuhn’s “Hungarian method” and some variants of shortest path algorithms, related to the computation of integers \(\ell _{i}\) such that a normal form may be obtained, in the generic case, by differentiating \(\ell _{i}\) times equation \(P_{i}\) . Fundamental results about changes of orderings and the various normal forms a system may have, including differential resolvents, are also provided. PubDate: 2022-03-25
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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: 2022-03-20
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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: 2022-03-03 DOI: 10.1007/s00200-022-00544-9
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Abstract: Let q be a prime power. PubDate: 2022-03-01 DOI: 10.1007/s00200-020-00445-9
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