8,044 research outputs found

    Integer linear programming techniques for constant dimension codes and related structures

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    The lattice of subspaces of a finite dimensional vector space over a finite field is combined with the so-called subspace distance or the injection distance a metric space. A subset of this metric space is called subspace code. If a subspace code contains solely elements, so-called codewords, with equal dimension, it is called constant dimension code, which is abbreviated as CDC. The minimum distance is the smallest pairwise distance of elements of a subspace code. In the case of a CDC, the minimum distance is equivalent to an upper bound on the dimension of the pairwise intersection of any two codewords. Subspace codes play a vital role in the context of random linear network coding, in which data is transmitted from a sender to multiple receivers such that participants of the communication forward random linear combinations of the data. The two main problems of subspace coding are the determination of the cardinality of largest subspace codes and the classification of subspace codes. Using integer linear programming techniques and symmetry, this thesis answers partially the questions above while focusing on CDCs. With the coset construction and the improved linkage construction, we state two general constructions, which improve on the best known lower bound of the cardinality in many cases. A well-structured CDC which is often used as building block for elaborate CDCs is the lifted maximum rank distance code, abbreviated as LMRD. We generalize known upper bounds for CDCs which contain an LMRD, the so-called LMRD bounds. This also provides a new method to extend an LMRD with additional codewords. This technique yields in sporadic cases best lower bounds on the cardinalities of largest CDCs. The improved linkage construction is used to construct an infinite series of CDCs whose cardinalities exceed the LMRD bound. Another construction which contains an LMRD together with an asymptotic analysis in this thesis restricts the ratio between best known lower bound and best known upper bound to at least 61.6% for all parameters. Furthermore, we compare known upper bounds and show new relations between them. This thesis describes also a computer-aided classification of largest binary CDCs in dimension eight, codeword dimension four, and minimum distance six. This is, for non-trivial parameters which in addition do not parametrize the special case of partial spreads, the third set of parameters of which the maximum cardinality is determined and the second set of parameters with a classification of all maximum codes. Provable, some symmetry groups cannot be automorphism groups of large CDCs. Additionally, we provide an algorithm which examines the set of all subgroups of a finite group for a given, with restrictions selectable, property. In the context of CDCs, this algorithm provides on the one hand a list of subgroups, which are eligible for automorphism groups of large codes and on the other hand codes having many symmetries which are found by this method can be enlarged in a postprocessing step. This yields a new largest code in the smallest open case, namely the situation of the binary analogue of the Fano plane

    Problems on q-Analogs in Coding Theory

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    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

    Bounds on List Decoding of Rank-Metric Codes

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    So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length nn and minimum rank distance dd. The third bound proves that there exists a rank-metric code over \Fqm of length nmn \leq m such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on nn and dd similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information Theory, short version presented at ISIT 201

    Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4

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    It is shown that the maximum size of a binary subspace code of packet length v=6v=6, minimum subspace distance d=4d=4, and constant dimension k=3k=3 is M=77M=77; in Finite Geometry terms, the maximum number of planes in PG(5,2)\operatorname{PG}(5,2) mutually intersecting in at most a point is 7777. Optimal binary (v,M,d;k)=(6,77,4;3)(v,M,d;k)=(6,77,4;3) subspace codes are classified into 55 isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any qq, yielding a new family of qq-ary (6,q6+2q2+2q+1,4;3)(6,q^6+2q^2+2q+1,4;3) subspace codes

    Equidistant Codes in the Grassmannian

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    Equidistant codes over vector spaces are considered. For kk-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker embedding, for 1-intersecting codes of kk-dimensional subspaces over \F_q^n, n(k+12)n \geq \binom{k+1}{2}, where the code size is qk+11q1\frac{q^{k+1}-1}{q-1} is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size n×(n2)n \times \binom{n}{2} over \F_q, rank n1n-1, and rank distance n1n-1.Comment: 16 page
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