Eurocode ’92: International Symposium on Coding Theory and by P. Camion, P. Charpin, S. Harari

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By P. Camion, P. Charpin, S. Harari

This publication is made up of the court cases of EUROCODE 1992 which used to be held in Udine (Italy) on the CISM, October 27 - 30, 1992. EUROCODE ’92 is a continuation in addition to an extension of the former colloquia Trois Journèes sur le codage and EUROCODE ’90, whose court cases seemed as Lecture Notes in machine technological know-how (Volumes 388 and 514). the purpose of EUROCODE ’92 used to be to draw excessive point learn papers and to motivate interchange of principles one of the parts of coding conception and similar fields which proportion a similar instruments for functions within the technology of communications, theoretical machine technological know-how, software program engineering and arithmetic. Then the ebook is characterised by way of a really large spectrum, starting from combinatories or algebraic geometry to implementation of coding algorithms. there have been approximately ninety members to the convention, from the educational and business worlds; fifty three meetings have been chosen. The submitted full-papers have been individually refereed for ebook through no less than foreign referees.

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Extra info for Eurocode ’92: International Symposium on Coding Theory and Applications

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The divisibility of Reed-Muller codes is known (Me El]. For every extended cyclic code can be located among the RM-codes, it has at least the divisibility of the smallest RM-code it is included in. We shall call this divisibility the minimum divisibility of the code. We recall the theorem of Mac Eliece, adapted here to the context of binary extended cyclic codes : Theorem 2 : Let C. be an extended cyclic code of length 2m over F 2 , with defining set T. Then, C. -divisible, where >. = w- 1, and r w = min{r, IT a"' = 1, u; E (0, n[\T} i=l a being a primitive root in F2m.

Proof. g. the all-zero word does not belong to C and there are exactly t codewords of C of weight 1. Suppose x is any other word of weight 1. Then x is covered by t codewords of C of weight 2 and none of them have any l's in common with the codewords of weight 1. • Example. If i is odd and t = (2i+ll/3, then n = 2i-t and K = t2n-i satisfy the equality K(l + n/t) = 2n. However, because n = 2t-l, no perfect binary {1,1/t)-covering of length n exists by the previous theorem. ACKNOWLEDGMENT The second author would like to thank Heikki Hamalainen and Markku Kaikkonen for useful discussions.

S. An antichain of (S, <::)is a subset of S of non-related elements. Let ~ be the map : ~:I C [O,n],..... ~(I)= Uteds E S, s <:: t} = Utei~({t}) A code of the algebra A (ie a F2-subspace of A) is called affine-invariant if its automorphism group contains the group GA(m) of affine permutations of F 2m : Pu,v : L: XgX 9 gEF2m --+ L: XgXug+v, u, V E f2m, U i- 0 gEF2m c. 1}) Let be an extended cyclic code of A, with defining set T •. Then, C. , or is an ideal of A. c. An elements is maximal (resp. minimal) in I if its strict ascendants (resp.

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