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By Jaques Calmet

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Extra resources for Algebraic Algorithms and Error-Correcting Codes: 3rd International Conference, AAECC-3 Grenoble, France, July 15–19, 1985 Proceedings

Example text

XP-1,---'-''''xP--1)',. This several authors gaves different riant under : P. D e l s a r t e s proofs the g r o u p of the of ali (17), fact A. that isometries Poli the o n l y are case (40) .... codes was which the g e n e r a l i z e d codes. b} studied These A. =. IFq _ i _Gl - . G. a. f. e. n. qroup, . . . . A. semi . . . simple. two are by authors inva- Reed-Muller 84 P. D e l s a r t e s simple gaves a c o m p l e t e survey in the case w h e r e A is semi (17) . F. minimal Mattson weight me idea, des uses isometries of q u a d r a t i c V.

U~ ; COMPLEXITY the process In t h a t case : W~ = U ; . T and W~ = M . U ~ I U/~ i s t h e e x p e c t e d E i g e n v e c t o r . (ROUGHESTIMATE) L e t S t h e number o f s t a t e s & M t h e o r d e r o f t h e code. Each s t e p r e q u i res S * M additions. The number o f s t e p s depends on t h e c o m p l e x i t y ( w e i g h t ) o4 t h e s o l u t i o n . An u p p e r bound i s M * S^2. The a v e r a g e number of increments per s t e p i s M ( e x p e r i m e n t a l ). T h i s g i v e s a complexity in 0 (M ~ S~:3) .

Various s t a t i s t i c a l p r o p e r t i e s o f t h e coded s i g n a l : Frequency spectrum (CARRIOLARO [ 2 ] , 3USTESEN [ 1 0 3 , LINDHOLM [123 ) , a b i l i t y t o s e l f s y n c h r o n i z i n g (DAVIS E 4 ] ) , e r r o r p r o p a g a t i o n d i s t r i b u t i o n (IMAI E g ] ) , e t c . . , + o l l o w t h e a b s o l u t e p r o b a b i l i t y o f STATES. Let the conditional probabilities of branches be t h e e l e m e n t s o+ a TRANSITION MATRIX. A w e l l known r e s u l t i s t h a t t h e p r o b a b i l i t y v e c t o r o f s t a t e s i s t h e M a t r i x EIGENVECTOR.

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