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Inequalities between the probability of a subspace and the probabilities of its cosets

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1 Author(s)

We consider an n -dimensional vector space over GF(q) which has a probability distribution defined on it. The sum of the probabilities over a proper k -dimensional subspace is compared to a sum over a coset of this subspace. The difference of these set probabilities is related to a sum of the Fourier transforms of the distribution over a subset of the domain of the transforms. We demonstrate the existence of a coset and both an upper and a lower bound on the difference associated with this coset. The bounds depend on the maximum and nonzero minimum of the transforms as defined on a special subset of the transform domain. Two examples from coding theory are presented. The first deals with a q -ary symmetric channel while the second is concerned with a binary compound channel.

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Information Theory, IEEE Transactions on  (Volume:19 ,  Issue: 4 )