New PDF release: A Survey of Algebraic Coding Theory: Lectures Held at the

By Elwyn R. Berlekamp

ISBN-10: 3211810889

ISBN-13: 9783211810880

ISBN-10: 370914325X

ISBN-13: 9783709143254

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Additional resources for A Survey of Algebraic Coding Theory: Lectures Held at the Department of Automation and Information, July 1970

Sample text

It is possible to select another basis which consists entirely of indicafor functions of (m- r) -dimensional affine subspaces. cu>+ 1) , _c<1) = _c + 1) may be replaced by """ which is the indicator Ml' function of an ( m- 2) -dimensional affine subspace. n order RM code is the smallest linear code contain- ing all such codewords.

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 . n, where u. is fixed while d. and n go to infinity), then we find that the number of information symbols increases as a power of I ( q, n, d) the block length. , then Berlekamp (1968, Chap.

The RM codes the include several important subclasses. For example, RM codes with r= m- 1 are the extended Hamming codes. If J is a p -ary descendant of L and if Cf then Wq(J) < 'W'q(i,) . From this it follows that every = pf, G-RM code is invariant under a little affine group. It is also easily seen that every codeword of the length qtn is also in the (r + 1)5 t ,-th q-ary G-RM code of order q -ary GRM code of length Cfm . RM code of next high er order. th order RM code but not in the ( r - 1)~ To keep the discussion simple, we first consider only the nary codes, for which N = lm.

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A Survey of Algebraic Coding Theory: Lectures Held at the Department of Automation and Information, July 1970 by Elwyn R. Berlekamp


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