By Elwyn R. Berlekamp

**Read Online or Download A Survey of Algebraic Coding Theory: Lectures Held at the Department of Automation and Information, July 1970 (CISM International Centre for Mechanical Sciences) PDF**

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**Extra resources for A Survey of Algebraic Coding Theory: Lectures Held at the Department of Automation and Information, July 1970 (CISM International Centre for Mechanical Sciences)**

**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.

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.

O~ N"