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IEEE Transactions on Information Theory

Bounds for the weight distribution of weakly self-dual codes

Vwani Roychowdhury, Farrokh Vatan

Published Jan 1, 2001• 5 citations
Publisher

Abstract

Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on quantum computing. This new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes. Moreover, in some cases, our bounds are improvements on the earlier bounds. These improvements are achieved either by extending the range of the weights over which the bounds apply or by extending the class of codes subjected to these bounds.

DOI
10.1109/18.904542

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Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on quantum computing. This new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes. Moreover, in some cases, our bounds are improvements on the earlier bounds. These improvements are achieved either by extending the range of the weights over which the bounds apply or by extending the class of codes subjected to these bounds.
Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on quantum computing. This new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes. Moreover, in some cases, our bounds are improvements on the earlier bounds. These improvements are achieved, either by extending the range of the weights over which the bounds apply, or by extending the class of codes subjected to these bounds.
In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.
Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory).
Using a variant of the linear programming method we derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives d/n/spl les/0.166315/spl middot//spl middot//spl middot/+o(1), thus improving on the Mallows-Odlyzko-Sloane bound of 1/6. To establish this, we prove that in any doubly even-self-dual code the distance distribution is asymptotically upper-bounded by the corresponding normalized binomial distribution in a certain interval.
For a given quaternary self-orthogonal (SO) code, using information of weights of codewords in the dual of the binary code generated by the supports of this SO code, one can construct new quaternary SO code by shortening method. Two methods of determining weights of codewords in the dual of the binary support code of a given large length SO code are presented. Using these methods, we construct many SO codes from a quantum 286-cap in PG(6, 4), and deduce existence of many quantum caps in PG(6, 4) and good quantum codes of distance 4.