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991.
We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and
designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code
has a large distance, then its dual has a small covering radius and, therefore, is large. This implies the original code to
be small.
We also point out that this bound is a natural isoperimetric constant of the Hamming cube, related to its Faber–Krahn minima.
While our approach belongs to the general framework of Delsarte’s linear programming method, its main technical ingredient
is Fourier duality for the Hamming cube. In particular, we do not deal directly with Delsarte’s linear program or orthogonal
polynomial theory.
This research was partially supported by ISF grant 039-7682. 相似文献
992.
We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients of some ideals in both number fields and cyclic algebras with a suitable trace form. We give criteria for unimodularity. This yields integral and unimodular lattices with a multiplicative structure. Examples are provided. 相似文献
993.
针对大气激光通信中低密度奇偶校验码(LDPC)置信传播(BP)译码算法复杂度高的问题,对几类BP-Based简化译码算法进行了分析,并基于最小均方误差准则(MMSE)对Scaled BP-Based和Offset BP-Based两类改进译码算法的优化设计进行了探讨,得出两类算法的最优校正因子,并给出了数值计算.在不同的湍流强度下,对码长1008的(6,3)比特填充LDPC码进行了仿真实验,结果表明,对于短码长的LDPC码,当译码BER=10-5时,最小和算法(UMP BP-Based)相对于BP算法有0.1~0.15dB的译码性能损失;基于MMSE设计的两类算法相比BP算法大大降低复杂度的同时,译码性能与BP算法相当,甚至优于BP算法,优于UMP BP-Based算法0.075~0.15dB. 相似文献
994.
In this note, we prove that if C is a duadic binary abelian code with splitting μ=μ−1 and the minimum odd weight of C satisfies d2−d+1≠n, then d(d−1)n+11. We show by an example that this bound is sharp. A series of open problems on this subject are proposed. 相似文献
995.
Let K(n,1) denote the minimal cardinality of a binary code of length n and covering radius one. Fundamental for the theory of lower bounds for K(n,1) is the covering excess method introduced by Johnson and van Wee. Let δi denote the covering excess on a sphere of radius i, 0≤i≤n. Generalizing an earlier result of van Wee, Habsieger and Honkala showed δp−1≥p−1 whenever n≡−1 (mod p) for an odd prime p and δ0=δ1=?=δp−2=0 holds. In the present paper we give the new estimation δp−1≥(p−2)p−1 instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ-function, which were conjectured by Habsieger. 相似文献
996.
For a k‐subset X of , the set of differences on X is the set (mod n): . A conflict‐avoiding code CAC of length n and weight k is a collection of k‐subsets of such that = ? for any distinct . Let CAC() be the class of all the CACs of length n and weight k. The maximum size of codes in CAC(n, k) is denoted by . A code CAC(n, k) is said to be optimal if = . An optimal code is tight equi‐difference if = and each codeword in is of the form . In this paper, the necessary and sufficient conditions for the existence problem of optimal tight equi‐difference conflict‐avoiding codes of length n = and weight 3 are given. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 223–231, 2013 相似文献
997.
998.
Explicit constructions in Extremal graph theory give appropriate lower bound for Turan type problems. In the case of prohibited cycles explicit constructions can be used in various problems of Information Security. We observe algebraic constructions of regular graphs of large girth and graphs with large cycle indicator and describe some algorithms of Coding Theory and Cryptography based on such special families of graphs. 相似文献
999.
We investigate the structure of codes over
rings with respect to the Rosenbloom-Tsfasman (RT) metric. We define a standard form generator matrix and show how we can
determine the minimum distance of a code by taking advantage of its standard form. We define MDR (maximum distance rank) codes
with respect to this metric and give the weights of the codewords of an MDR code. We explore the structure of cyclic codes
over
and show that all cyclic codes over
rings are MDR. We propose a decoding algorithm for linear codes over these rings with respect to the RT metric.
AMS Classification: 94B05, 94B60 相似文献
1000.
Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with
For n = 2 or 3 the characteristic function
of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of
is
for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg(
for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least
相似文献