Download PDF by Elwyn R. Berlekamp: 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

For all primitive binary BC. ~= = 121, d. 8050 = 2 9 , 31 . The Kasami-Tokura counterexamples are based on the following result, which we state without proof : BCH Bound 57 McEliece's Theorem : . If K does not contain any j--tuple, x. TI-K(Z - ~ ~) is a multiple of Corollary 1 : length n liE. zi. All weights in the expurgated binary QR code of =-1 mod. 8 are divisible by 4. Proof : In this case, -1 is a nonresidue, the negative or eve- ry residue is a nonresidue, and no pair of residues (or non residues )can sum to Corollary 2 : 0 .

The null space does not contain the indicator functions of the dimensional affine subs paces. F(Zm) r-dimensional affine subspace of and let Q (5) be its indicator function. The entire space of m-dimensional binary vectors may be partitioned into zm-r disjoint sets, each of which is a translate of S. , .. , Tj Q(T1 ) .. , , Q(Ti) 1 translates which are distinct from S by and their indicator functions by Q(T~) , . nslates, S U T~ is an ( r + 1) -dimensional affine subspace. It follows that Q(S) + Q(Ti,) is in the null space of the rth order RM code.

If a different ex. is chosen, one gets an equivalent(but not identical)code. +i a e R0 . , merely represents a change of the primitive root , OG. ___.. tA. +d. ,veGF(tt)U{oo} . The projective unimodular subgroup of the linea:r fractional group is transitive (in fact, it is doubly transitive), so we may apply Prange's Theorem to conclude that the minimum nonzero weight of every binary QR code is odd. i) mod. "-1. ) C. ) tiple of . We then observe that since is a multiple of . nR( X. n ~e (X R1 l.

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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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