2,148 research outputs found

    Problems on q-Analogs in Coding Theory

    Full text link
    The interest in qq-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

    On the lengths of divisible codes

    Get PDF
    In this article, the effective lengths of all qrq^r-divisible linear codes over Fq\mathbb{F}_q with a non-negative integer rr are determined. For that purpose, the Sq(r)S_q(r)-adic expansion of an integer nn is introduced. It is shown that there exists a qrq^r-divisible Fq\mathbb{F}_q-linear code of effective length nn if and only if the leading coefficient of the Sq(r)S_q(r)-adic expansion of nn is non-negative. Furthermore, the maximum weight of a qrq^r-divisible code of effective length nn is at most σqr\sigma q^r, where σ\sigma denotes the cross-sum of the Sq(r)S_q(r)-adic expansion of nn. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace codes

    Covering of Subspaces by Subspaces

    Full text link
    Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph \cG_q(n,k), k≥rk \geq r, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, qq-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for q=2q=2 with r=2r=2 or r=3r=3. We discuss the density for some of these coverings. Tables for the best known coverings, for q=2q=2 and 5≤n≤105 \leq n \leq 10, are presented. We present some questions concerning possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352

    Characterisations of elementary pseudo-caps and good eggs

    Get PDF
    In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in PG(4n−1,q)\mathrm{PG}(4n-1, q), qq odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle T3(O)T_3(\mathcal{O}) of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw

    Identifying Infection Sources and Regions in Large Networks

    Full text link
    Identifying the infection sources in a network, including the index cases that introduce a contagious disease into a population network, the servers that inject a computer virus into a computer network, or the individuals who started a rumor in a social network, plays a critical role in limiting the damage caused by the infection through timely quarantine of the sources. We consider the problem of estimating the infection sources and the infection regions (subsets of nodes infected by each source) in a network, based only on knowledge of which nodes are infected and their connections, and when the number of sources is unknown a priori. We derive estimators for the infection sources and their infection regions based on approximations of the infection sequences count. We prove that if there are at most two infection sources in a geometric tree, our estimator identifies the true source or sources with probability going to one as the number of infected nodes increases. When there are more than two infection sources, and when the maximum possible number of infection sources is known, we propose an algorithm with quadratic complexity to estimate the actual number and identities of the infection sources. Simulations on various kinds of networks, including tree networks, small-world networks and real world power grid networks, and tests on two real data sets are provided to verify the performance of our estimators
    • …
    corecore