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By Elwyn R. Berlekamp

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Since multiplication of a number by q mod,qm-1 is merely a cyclic shift of them-digit q -ary expansion of that number, it is natural to represent the elements of K in terms of their m- digit q- ary expansions whenever n = q'"- 1 . The condition that qK = R merely requires that the cyclic shifts of them-dig- it q-ary representation of each number in R must be another number inK. If J is a number relatively prime to qm-1 , then the permutation u,-u! in G-F(qm) permutes each extend- ed cyclic code into an equivalent extended cyclic code.

In face, the digits of such a code can still be directly associated with the elements of '-FC pm) in such a way as to leave the code invariant under the little affine group. If one associates the successive digits of the code with the successive powers of a wrong primitive element, then the code is not invariant under the translational group because the addition in G-F (qm) is not correctly defined. In other words, the previous theorem says only that if R does not contain all of its own descendants, then the extended code is not invariant under the little affine group if the digits of the code are associated with the elements of G-F (qm) in a particular manner.

805 e + 1 in certain cases. These conditions, too complicated to be presented here, were later refined by Chien and Lum (1966). Hartmann and Tzeng (1970) obtained separate bounds for the minimum odd weight and the minimum nonzero even weight. A summary and these and other results on the minimum distances of imprimitive BCH codes is given by Hartmann (1970). The rates and distances of long BC. H codes 60 BCH Bound turn out to be related in a very interesting way. If we consider a sequence of BCHcodes of increasing block lengths in which the distances .

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