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Register a new userOn tensor products of CSS Codes
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Added by
Benjamin Audoux
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product $otimes$ which induces a similar operation on the former. We investigate this operation, and in particular its behavior with regard to minimum distances. Given a CSS code $mathcal{C}$, we give a criterion which provides a lower bound on the minimum distance of $mathcal{C} otimes mathcal{D}$ for every CSS code $mathcal D$. We apply this result to study the behaviour of iterated tensor powers of codes. Such sequences of codes are logarithmically LDPC and we prove in particular that their minimum distances tend generically to infinity. Different known results are reinterpretated in terms of tensor products. Three new families of CSS codes are defined, and their iterated tensor powers produce LDPC sequences of codes with length $n$, row weight in $O(log n)$ and minimum distances larger than $n^{frac{alpha}{2}}$ for any $alpha<1$. One family produces sequences with dimensions larger than $n^beta$ for any $beta<1$.
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We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear $2$-weight code with $delta > 1$ implies the following equality of great common divisors: $(d,q) = (delta,q)$. Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small $q = 2,3,4$ and $q,n < 50$ are presented.
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