Authors:Yun Gao; Jian Gao; Tingting Wu; Fang-Wei Fu Pages: 457 - 467 Abstract: In this paper, we study 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring \(R=\frac{{{\mathbb {Z}_4}[u]}}{{\left\langle {{u^2} - 1} \right\rangle }}\) . We determine the structure of the generators and the minimal generating sets of 1-generator QC and GQC codes. We also give a lower bound for the minimum distance of free 1-generator quasi-cyclic and generalized quasi-cyclic codes over this ring, respectively. Furthermore, some new \(\mathbb {Z}_4\) -linear codes via the Gray map which have better parameters than the best known \(\mathbb {Z}_4\) -linear codes are presented. PubDate: 2017-12-01 DOI: 10.1007/s00200-017-0315-1 Issue No:Vol. 28, No. 6 (2017)

Authors:Zohreh Rajabi; Kazem Khashyarmanesh Pages: 469 - 495 Abstract: Cyclic codes are an important class of linear codes. The objectives of this paper are to earn and extend earlier results over cyclic codes from some monomials. In fact, we determine the dimension and the generator polynomial of the code \({\mathcal {C}}_s\) defined by the monomial \(f(x)=x^{\frac{p^h+1}{2}}\) over \({\mathrm {GF}}(p^m)\) , where p is an odd prime and h is an integer. Also, we provide some answers for Open Problems 5.26 and 5.30 in Ding (SIAM J Discrete Math 27:1977–1994, 2013). Moreover, we study the code \({\mathcal {C}}_s\) defined by the monomial \(f(x)=x^{\frac{q^h-1}{q-1}}\) over \(\mathrm {GF}(q^m)\) , where h is an integer, without any restriction on h (see Section 5.3 in the above mentioned paper). PubDate: 2017-12-01 DOI: 10.1007/s00200-017-0317-z Issue No:Vol. 28, No. 6 (2017)

Authors:Krzysztof Ziemiański Pages: 497 - 525 Abstract: The spaces of directed paths on the geometric realizations of pre-cubical sets, called also \(\square \) -sets, can be interpreted as the spaces of possible executions of Higher Dimensional Automata, which are models for concurrent computations. In this paper we construct, for a sufficiently good pre-cubical set K, a CW-complex \(W(K)_v^w\) that is homotopy equivalent to the space of directed paths between given vertices v, w of K. This construction is functorial with respect to K, and minimal among all functorial constructions. Furthermore, explicit formulas for incidence numbers of the cells of \(W(K)_v^w\) are provided. PubDate: 2017-12-01 DOI: 10.1007/s00200-017-0316-0 Issue No:Vol. 28, No. 6 (2017)

Authors:Gerardo Vega Pages: 527 - 533 Abstract: The purpose of this work is to use an already known identity among the weight enumerator polynomials, in order to present an improved method for determining the weight distribution of a family of q-ary reducible cyclic codes, that generalize, in an easier way, the results in Yu and Liu (Des Codes Cryptogr 78:731–745, 2016). PubDate: 2017-12-01 DOI: 10.1007/s00200-017-0318-y Issue No:Vol. 28, No. 6 (2017)

Authors:Xiaoni Du; Yunqi Wan Pages: 535 - 547 Abstract: Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power q, we present a class of linear codes over finite fields \(F_q\) with quadratic forms via a general construction and then determine the explicit complete weight enumerators of these linear codes. Our construction covers some related ones via quadratic form functions and the linear codes may have applications in cryptography and secret sharing schemes. PubDate: 2017-12-01 DOI: 10.1007/s00200-017-0319-x Issue No:Vol. 28, No. 6 (2017)

Authors:Michele Rossi; Lea Terracini Pages: 351 - 368 Abstract: The main object of the present paper is a numerical criterion determining when a Weil divisor of a \({\mathbb {Q}}\) –factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of \({\mathbb {Z}}\) –linear interpretation of Gale duality and secondary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak \({\mathbb {Q}}\) –Fano, \({\mathbb {Q}}\) –Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied. PubDate: 2017-11-01 DOI: 10.1007/s00200-016-0308-5 Issue No:Vol. 28, No. 5 (2017)

Authors:W. Fish Pages: 369 - 386 Abstract: Let \(n, m \ge 2\) be integers. The cartesian, categorical and lexicographic products of m copies of the n-cycle denoted by \(C_n\) all have as their vertex-set \(\{0, 1, \ldots , n-1\}^m\) , with adjacency defined variously. In this paper the binary codes generated by the row span of adjacency matrices of the cartesian, categorical and lexicographic products of m copies of \(C_n\) are examined. Full and partial PD-sets were also found for the various codes. PubDate: 2017-11-01 DOI: 10.1007/s00200-016-0310-y Issue No:Vol. 28, No. 5 (2017)

Authors:Miriam Abdón; Robert Rolland Pages: 387 - 408 Abstract: For any finite field \({\mathbb {F}}_q\) with q elements, we study the set \({\mathscr {F}}_{(q,m)}\) of functions from \({\mathbb {F}}_q^m\) into \({\mathbb {F}}_q\) from geometric, analytic and algorithmic points of view. We determine a linear system of \(q^{m+1}\) equations and \(q^{m+1}\) unknowns, which has for unique solution the Hamming distances of a function in \({\mathscr {F}}_{(q,m)}\) to all the affine functions. Moreover, we introduce a Fourier-like transform which allows us to compute all these distances at a cost \(O(mq^m)\) and which would be useful for further problems. PubDate: 2017-11-01 DOI: 10.1007/s00200-016-0311-x Issue No:Vol. 28, No. 5 (2017)

Authors:T. Aaron Gulliver; Masaaki Harada Pages: 409 - 424 Abstract: We study the performance of ternary isodual codes which are not self-dual and ternary self-dual codes, as measured by the decoding error probability with bounded distance decoding. We compare the performance of ternary double circulant and double twistulant codes which are not self-dual with ternary extremal self-dual codes. We also investigate the performance of ternary self-dual codes having large minimum weights. PubDate: 2017-11-01 DOI: 10.1007/s00200-017-0312-4 Issue No:Vol. 28, No. 5 (2017)

Authors:Riccardo Aragona; Riccardo Longo; Massimiliano Sala Pages: 425 - 436 Abstract: In this paper we propose a tokenization algorithm of Reversible Hybrid type, as defined in PCI DSS guidelines for designing a tokenization solution, based on a block cipher with a secret key and (possibly public) additional input. We provide some formal proofs of security for it, which imply our algorithm satisfies the most significant security requirements described in PCI DSS tokenization guidelines. Finally, we give an instantiation with concrete cryptographic primitives and fixed length of the PAN, and we analyze its efficiency and security. PubDate: 2017-11-01 DOI: 10.1007/s00200-017-0313-3 Issue No:Vol. 28, No. 5 (2017)

Authors:Ulrich Oberst Pages: 437 - 456 Abstract: We complete the stability results of the paper Bourlès et al. (SIAM J Control Optim 53:2725–2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators. PubDate: 2017-11-01 DOI: 10.1007/s00200-017-0314-2 Issue No:Vol. 28, No. 5 (2017)

Authors:Pamela E. Harris; Erik Insko; Anthony Simpson Abstract: The multiplicity of a weight \(\mu \) in an irreducible representation of a simple Lie algebra \(\mathfrak {g}\) with highest weight \(\lambda \) can be computed via the use of Kostant’s weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a q-analog of Kostant’s weight multiplicity and present a SageMath program to compute q-multiplicities for the simple Lie algebras. PubDate: 2017-11-01 DOI: 10.1007/s00200-017-0346-7

Authors:Minjia Shi; Liqin Qian; Patrick Solé Abstract: In this paper, few weights linear codes over the local ring \(R={\mathbb {F}}_p+u{\mathbb {F}}_p+v{\mathbb {F}}_p+uv{\mathbb {F}}_p,\) with \(u^2=v^2=0, uv=vu,\) are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to \({\mathbb {F}}_p^4,\) we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and \(p\equiv 3 ~(\mathrm{mod} ~4).\) Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented. PubDate: 2017-10-24 DOI: 10.1007/s00200-017-0345-8

Authors:S. Pumplün Abstract: We show how cyclic \((f,\sigma ,\delta )\) -codes over finite rings canonically induce a \({\mathbb {Z}}\) -lattice in \({\mathbb {R}}^N\) by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain \(\sigma \) -constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases. PubDate: 2017-10-16 DOI: 10.1007/s00200-017-0344-9

Authors:Pınar Çomak; Jon Lark Kim; Ferruh Özbudak Abstract: We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length \(3\ell \) with the algebraic approach of Ling and Solé (IEEE Trans Inf Theory 47(7):2751–2760, 2001. doi:10.1109/18.959257). In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66. PubDate: 2017-09-25 DOI: 10.1007/s00200-017-0343-x

Authors:Bentolhoda Binaei; Amir Hashemi; Werner M. Seiler Abstract: In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bézout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem. PubDate: 2017-09-15 DOI: 10.1007/s00200-017-0342-y

Authors:Dae-Woong Lee Abstract: In this paper, we study the digital Hopf groups and the digital Hopf functions between digital Hopf spaces with digital multiplications, and construct a near-ring structure on the set of all pointed digital homotopy classes of digital Hopf functions between pointed digital Hopf groups. We also investigate a near-ring homomorphism between near-rings based on the pointed digital Hopf groups to find a new method of how to give answers to the original problems or how to get a new information out of old ones more effectively. PubDate: 2017-09-14 DOI: 10.1007/s00200-017-0341-z

Authors:Omar Akchiche; Omar Khadir Abstract: We address the problem of factoring a large RSA modulus \(N=pq\) with p and q sharing a portion of bits in the middle. New polynomial time algorithms for computing the prime decomposition of N under certain conditions are presented. As an application, several attacks against RSA system using this class of moduli with low public exponent are described. Our results suggest that such integers are not appropriate for cryptographic purposes. PubDate: 2017-08-21 DOI: 10.1007/s00200-017-0340-0

Authors:J. D. Key; B. G. Rodrigues Abstract: It is shown how LCD codes with a particularly useful feature can be found from row spans over finite fields of adjacency matrices of graphs by considering these together with the codes from the associated reflexive graphs and complementary graphs. Application is made to some particular classes, including uniform subset graphs and strongly regular graphs where, if a p-ary code from a graph has this special LCD feature, the dimension can be found from the multiplicities modulo p of the eigenvalues of an adjacency matrix and, bounds on the minimum weight of the code and the dual code follow from the valency of the graph. PubDate: 2017-08-19 DOI: 10.1007/s00200-017-0339-6

Authors:Can Xiang; Xianfang Wang; Chunming Tang; Fangwei Fu Abstract: Linear codes have been an interesting topic in both theory and practice for many years. In this paper, two classes of linear codes over the finite field \({\mathrm {GF}}(p)\) are presented and their weight distributions are also determined, where p is an odd prime. Some of the linear codes obtained are optimal or almost optimal in the sense that their parameters meet certain bound on linear codes. PubDate: 2017-08-18 DOI: 10.1007/s00200-017-0338-7