Coding and information theory
Designs, Codes and Cryptography
Algebraic-Geometric Codes
Introduction to Coding Theory
A characterization of codes with extreme parameters
IEEE Transactions on Information Theory - Part 2
Extended Binomial Moments of a Linear Code and the Undetected Error Probability
Problems of Information Transmission
Discrete Applied Mathematics
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The parameters of a linear codeC over GF(q) are given by [n,k,d],where n denotes the length, k the dimensionand d the minimum distance of C. Thecode C is called MDS, or maximum distance separable,if the minimum distance d meets the Singleton bound,i.e. d = n-k+1. Unfortunately, the parameters ofan MDS code are severely limited by the size of the field. Thuswe look for codes which have minimum distance close to the Singletonbound. Of particular interest is the class of almost MDS codes,i.e. codes for which d=n-k. We will present a conditionon the minimum distance of a code to guarantee that the orthogonalcode is an almost MDS code. This extends a result of Dodunekovand Landgev Dodunekov. Evaluation of the MacWilliams identitiesleads to a closed formula for the weight distribution which turnsout to be completely determined for almost MDS codes up to oneparameter. As a consequence we obtain surprising combinatorialrelations in such codes. This leads, among other things, to ananswer to a question of Assmus and Mattson 5 on the existenceof self-dual [2d,d,d] -codes which have no codewords of weight d+1 . Actually there are more codesthan Assmus and Mattson expected, but the examples which we knoware related to the expected ones.