On the Clifford defect for special curves

Kirfel, Christoph

doi:10.1515/9783110811056.67

Cited by 2 publications

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“…It is shown in [11,9,15] that the FR bound improves the Goppa bound for algebraic-geometric codes. Furthermore we gave several properties for the FR bound in terms of the semigroup of non-gaps, see [14,15]. Majority coset decoding decodes up to half the FR bound.…”

Section: Theorem 54mentioning

confidence: 99%

The shift bound for cyclic, Reed-Muller and geometric Goppa codes

Pellikaan

Arithmetic, Geometry, and Coding Theory

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We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed-Muller and algebraicgeometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound.1991 Mathematics Subject Classification: 94B27, 14H45. IntroductionIn this paper we survey various bounds on the minimum distance of cyclic and algebraic-geometric codes. For cyclic codes the BCH, Hartmann-Tzeng [12,27], Roos [22] and the shift bound of van Lint, Wilson and van Eupen [7,16] for the minimum distance are well known. In Section 2 we give several formulations of independent sets for the definition of the shift bound for cyclic codes. With this last reformulation we obtained, in cooperation with Shen and Tzeng [25], a generalization of the definition of independent sets 1 The shift bound 2 and the shift bound for arbitrary linear codes in Section 3. It appears that all these bounds give a method for finding non-singular square submatrices of a matrix of syndromes. The shift bound for Reed-Muller codes is equal to the minimum distance. In Section 4 we treat shift bounds on generalized weights. Error-correcting arrays [15,20] are treated in Section 5. With this concept the Feng-Rao bound [11,9] is generalized from geometric Goppa codes to arbitrary linear codes. In this setting one gets in general an improvement of the Goppa bound of algebraic-geometric codes. Although it is in many cases the true minimum distance, this is not always the case. In Section 6 we sketch how majority coset decoding of Feng-Rao [8] and Duursma [3,4] corrects up to half the shift bound. This procedure is in the worst case not efficient. Therefore we define a restricted shift bound such that the proposed algorithm has polynomial complexity. The shift bound improves all bounds except the Roos bound. The restricted shift bound still improves the FR bound and is equal to the minimum distance of Reed-Muller codes.A finite field is denoted by F and the multiplicative group of non-zero elements by F * . We denote a subfield of F by F 0 . The finite field with q elements is denoted by F q . We denote the coordinatewise multiplication of a and b in F n by a * b, so a i b i is the ith coordinate of a * b. With this multiplication F n becomes an F-algebra. We define < a, b >= a i b i . The integers are denoted by Z, the positive integers are denoted by N and the non-negative integers by N 0 . The integers modulo n are denoted by Z n . The number of elements of a finite set A we denote by #A. Cyclic codes and the shift boundFundamental for the definition of the shift bound is the notion of an independent set. In the sequel we give several equivalent definitions of this notion for cyclic codes. In Section 3 we generalize one of these definitions to a broader context which is well suited for Reed-Muller and algebraic-geometric codes.Definition 2.1. Let F be a finite field. Let α be an element in F * of order n. Let J be a subset of Z n . Define the F-linear codeC(J) bỹand theThe shift bound 3 C(J...

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