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A Survey of Algebraic Coding Theory: Lectures Held at the by Elwyn R. Berlekamp

By Elwyn R. Berlekamp

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

Sample text

N- 1 L=O But the weight of the product cannot excede the product of the weights, so we deduce that the minimum weight, d, , of the augmented binary Q R code of length inequality cl 2. > n satisfies the n . We have thus shown that the extended binary QR codes, each of which has rate distances approaching e><> R= 1/2 , have minimum with increasing block length, although the bound is disappointingly weak. The Gilbert bound assures us that there exist arbitrarily long codes of rate 1/2 whose distances grow linearly with block lengths.

For small u. is really quite remarkable. Although s' (IN)= 0 almost all positive ""', s'(O) = -1/ Ln 4 . for 63 Algebraic Decoding of BCH Codes The approximation d, ~zn lnR-Ylogl1 shows that the distances of long BCH codes are much better than RM codes or the best lower bounds known on the distances of QR codes. H codes grows not as Vn, but as n \n 4/ tog n . 9. Algebraic Decoding of BCH Codes. H codes is the existence of an easily implement- ed decoding algorithm we shall now describe. The goal of the decoder is to determine the error polynomial, E (X) = n-1 r: ~=0 .

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. n- 1 L=O But the weight of the product cannot excede the product of the weights, so we deduce that the minimum weight, d, , of the augmented binary Q R code of length inequality cl 2.

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