Download PDF by Jacobus H. van Lint (auth.): Coding Theory

By Jacobus H. van Lint (auth.)

ISBN-10: 3540054766

ISBN-13: 9783540054764

ISBN-10: 3662207125

ISBN-13: 9783662207123

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Sample text

These are code words 1n M~+ Mj. a _II Furthermore per(T ! + 'I:£) • H. JJ. ~ + Finally, suppose T £ a' and T ! + b are in tho 54 same cycle of M~ + Mj. 10)~: "ee' (mod h). TA(Ta~ + ~) _ Ta' a + £, Le. :. :. ; + £ (0 :::: ex < h) are representatives. d (or COClputer for slightly larger codes). Take n • 15. 6) we found the five primitive idempotents to be ~1 :-::Je ~ :.. L. 1.. x 3 +x 6 +x 7 +x9+ x 11 +x'2+ x'3+ x I 4 c +~"'x+x 2 3 4 6 7 8 9 11 12 13 14 x+x +x +x +x +x +x +x +x +x +x +x 2 3 4 6 8 Q +x +x +x +x +x"'+x 12 2 4 5 7 8 10 '1 13 14 ~ :..

O j *k i ) = O. 49 Therefore there is a permutation which permutes all code words of ~g(x) into code words of ~g*(x). The important thing we learn from the poi:1t of vi .... of this paragraph is that the cyclic shifts are not the only pemutations which leave a cyclic code invariant. g. we take a bir~ry If cyclic code of odd word length n then the permutation n2 per- mutes code words into code words. t enumerators for cyclic Codes. 3). The larger the automorphimn group of a code is tie easier it is in general to find the weight enumerator of the code.

5) is it is easily carried out by computer. tl~t We are nOW' in a position to enumerate the weights of code words in a cyclic code. In the following T denotes the permutation k .. k + 1 (IOOd n) or the mapping of R given by T(g(x» ~ xg(x}. If a(x) is a code word 1n a cyclic code then 2 n-l T~ T ~ ••• , T a are also code words in this code but not necessarily all different. ). Let a(x} E R. _ (x n _l}/g{x). ) = e. Proof: (1) Define g*(x) :c (x e _l)/h(x). (c(x},xn_l) c I. Let a(x} c c(x}g(x) witr.

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Coding Theory by Jacobus H. van Lint (auth.)

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