By V. B. Balakirsky (auth.), G. Cohen, S. Litsyn, A. Lobstein, G. Zémor (eds.)

This quantity offers the lawsuits of the 1st French-Israeli Workshop on Algebraic Coding, which came about in Paris in July 1993. The workshop was once a continuation of a French-Soviet Workshop held in 1991 and edited by means of an analogous board. The completely refereed papers during this quantity are grouped into components on: convolutional codes and specific channels, overlaying codes, cryptography, sequences, graphs and codes, sphere packings and lattices, and limits for codes.

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Z nq I q . We therefore have epimorphisms F V /Q F V /I and F V /Q q F V /I q . 1 Q : I is principal over Q, say Q : I = u + Q. 3 Q q : I q = u q + Q q . 5 we learn that s s ° z q −1 · · · z q −1 uzD = n 1 48 MACAULAY’S DUAL SYSTEMS AND FROBENIUS POWERS [PART II ° b denotes that a and b are nonzero scalar multiin F V Q, where a = ples of each other. Without loss of generality we may suppose u z D and s s z1q −1 · · · z nq −1 are equal. 3 we conclude that z q D z1q−1 · · · z nq−1 represents a fundamental class for F V /I q .

Z n , where as before m F z1 , . . , z n is the augmentation ideal z1 , . . , z n . One deﬁnition of e1 , . . , e n is via the identity n er X s , X + zi = r+s=n i=1 with the convention that e0 = 1. Note that the left hand side is obviously zero for X = −z i , i = 1 , . . , n. If we write the resulting n equations as a single matrix equation and rearrange terms we ﬁnd −1 n −z1n . . −z nn Therefore z1n , . . , z nn = −z1 . . −z n n−1 n−1 · · · −z1 . . . · · · −z n 1 . . 1 e1 , . . , e n , so z1n , .

2 I F V is an F V -primary irreducible ideal. 1, one can prove this directly, as I consists of all elements of F V for which there is no Poincar´e dual element in the quotient algebra F V I . The following result in different language is due to F. S. Macaulay ([49] Section 72). 1 (F. S. Macaulay): Let F be a ﬁeld, n N , and set V = F n . The assignment I induces a correspondence between nonzero elements V d and F V -primary irreducible ideals I F V such that the corresponding Poincar´e duality quotient algebra F V I has formal V d determine the same dimension d .