13,147 research outputs found
On squares of cyclic codes
The square of a linear error correcting code is the linear code
spanned by the component-wise products of every pair of (non-necessarily
distinct) words in . Squares of codes have gained attention for several
applications mainly in the area of cryptography, and typically in those
applications one is concerned about some of the parameters (dimension, minimum
distance) of both and . In this paper, motivated mostly by the
study of this problem in the case of linear codes defined over the binary
field, squares of cyclic codes are considered. General results on the minimum
distance of the squares of cyclic codes are obtained and constructions of
cyclic codes with relatively large dimension of and minimum distance of
the square are discussed. In some cases, the constructions lead to
codes such that both and simultaneously have the largest
possible minimum distances for their length and dimensions.Comment: Accepted at IEEE Transactions on Information Theory. IEEE early
access version available at https://ieeexplore.ieee.org/document/8451926
Low-complexity quantum codes designed via codeword-stabilized framework
We consider design of the quantum stabilizer codes via a two-step,
low-complexity approach based on the framework of codeword-stabilized (CWS)
codes. In this framework, each quantum CWS code can be specified by a graph and
a binary code. For codes that can be obtained from a given graph, we give
several upper bounds on the distance of a generic (additive or non-additive)
CWS code, and the lower Gilbert-Varshamov bound for the existence of additive
CWS codes. We also consider additive cyclic CWS codes and show that these codes
correspond to a previously unexplored class of single-generator cyclic
stabilizer codes. We present several families of simple stabilizer codes with
relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl
Quantum Synchronizable Codes From Quadratic Residue Codes and Their Supercodes
Quantum synchronizable codes are quantum error-correcting codes designed to
correct the effects of both quantum noise and block synchronization errors.
While it is known that quantum synchronizable codes can be constructed from
cyclic codes that satisfy special properties, only a few classes of cyclic
codes have been proved to give promising quantum synchronizable codes. In this
paper, using quadratic residue codes and their supercodes, we give a simple
construction for quantum synchronizable codes whose synchronization
capabilities attain the upper bound. The method is applicable to cyclic codes
of prime length
A Complete Characterization of Irreducible Cyclic Orbit Codes
We give a complete list of orbit codes that are generated by an irreducible
cyclic group, i.e. an irreducible group having one generator. We derive some of
the basic properties of these codes such as the cardinality and the minimum
distance.Comment: in Proceedings of The Seventh International Workshop on Coding and
Cryptography 2011 April 11-15 2011, Paris, Franc
- …