443 research outputs found
A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field
Using the Weyl commutation relations over a finite field we introduce a
family of error-correcting quantum stabilizer codes based on a class of
symmetric matrices over the finite field satisfying certain natural conditions.
When the field is GF(2) the existence of a rich class of such symmetric
matrices is demonstrated by a simple probabilistic argument depending on the
Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these
symmetric matrices are assumed to be circulant it is possible to obtain
concrete examples by a computer program. The quantum codes thus obtained admit
elegant encoding circuits.Comment: 16 pages, 2 figure
Constructions of Abelian Codes multiplying dimension of cyclic codes
In this note, we apply some techniques developed in [1]-[3] to give a
particular construction of bivariate Abelian Codes from cyclic codes,
multiplying their dimension and preserving their apparent distance. We show
that, in the case of cyclic codes whose maximum BCH bound equals its minimum
distance the obtained abelian code verifies the same property; that is, the
strong apparent distance and the minimum distance coincide. We finally use this
construction to multiply Reed-Solomon codes to abelian codesComment: arXiv admin note: text overlap with arXiv:2402.0393
Quantum two-block group algebra codes
We consider quantum two-block group algebra (2BGA) codes, a previously
unstudied family of smallest lifted-product (LP) codes. These codes are related
to generalized-bicycle (GB) codes, except a cyclic group is replaced with an
arbitrary finite group, generally non-abelian. As special cases, 2BGA codes
include a subset of square-matrix LP codes over abelian groups, including
quasi-cyclic codes, and all square-matrix hypergraph-product codes constructed
from a pair of classical group codes. We establish criteria for permutation
equivalence of 2BGA codes and give bounds for their parameters, both explicit
and in relation to other quantum and classical codes. We also enumerate the
optimal parameters of all inequivalent connected 2BGA codes with stabilizer
generator weights , of length for abelian groups, and for non-abelian groups.Comment: 19 pages, 9 figures, 3 table
50 Years of the Golomb--Welch Conjecture
Since 1968, when the Golomb--Welch conjecture was raised, it has become the
main motive power behind the progress in the area of the perfect Lee codes.
Although there is a vast literature on the topic and it is widely believed to
be true, this conjecture is far from being solved. In this paper, we provide a
survey of papers on the Golomb--Welch conjecture. Further, new results on
Golomb--Welch conjecture dealing with perfect Lee codes of large radii are
presented. Algebraic ways of tackling the conjecture in the future are
discussed as well. Finally, a brief survey of research inspired by the
conjecture is given.Comment: 28 pages, 2 figure
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