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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

And N are large, then most error patterns of weight slightly more than d/2 will not be evenly distributed among the translates of any r-dimensional subspace S. zm- '"- 1 Less than half of these translates will contain an odd number of errors and the elections will determine the correct values of Q(S)Et for all r-dimensional subspaces S. > \I'N. However, this lower bound on the minimum distance of QR codes appears to be weak, and in all known cases the QR. codes are actually as good as or better than the RM codes, when goodness is measured in terms of minimum distance.

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.

Zm+ 1 t ams vectors. _ al affine subspaces. The first order RMcode is the orthogonal complement of the extended Hamming code. The generator matrix of the first order RMcode may be written in the following way, 43 The Affine Groups 1 ... 1 ] OGio •• , OGN-2 where each of the last m rows is them -dimensional binary vector obtained by representing each element of G-F(lm) in terms of some m-dimensional basis over '- F(l). ,_, der code. • , XNYN] • We now assert that for ~ - """ any choice of the binary variables Ah A1 , A3 , •..

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