60,492 research outputs found
Quantum MDS Codes over Small Fields
We consider quantum MDS (QMDS) codes for quantum systems of dimension
with lengths up to and minimum distances up to . We show how
starting from QMDS codes of length based on cyclic and constacyclic
codes, new QMDS codes can be obtained by shortening. We provide numerical
evidence for our conjecture that almost all admissible lengths, from a lower
bound on, are achievable by shortening. Some additional codes that
fill gaps in the list of achievable lengths are presented as well along with a
construction of a family of QMDS codes of length , where , that
appears to be new.Comment: 6 pages, 3 figure
The q-ary image of some qm-ary cyclic codes: permutation group and soft-decision decoding
Using a particular construction of generator matrices of
the q-ary image of qm-ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups. The equivalence of such codes with some two-dimensional (2-D) Abelian codes and cyclic codes is deduced from this property. These permutations are also used in the area of the soft-decision decoding of some expanded Reed–Solomon (RS) codes to improve the performance of generalized minimum-distance decoding
Universal Gr\"obner Bases for Binary Linear Codes
Each linear code can be described by a code ideal given as the sum of a toric
ideal and a non-prime ideal. In this way, several concepts from the theory of
toric ideals can be translated into the setting of code ideals. It will be
shown that after adjusting some of these concepts, the same inclusion
relationship between the set of circuits, the universal Gr\"obner basis and the
Graver basis holds. Furthermore, in the case of binary linear codes, the
universal Gr\"obner basis will consist of all binomials which correspond to
codewords that satisfy the Singleton bound and a particular rank condition.
This will give rise to a new class of binary linear codes denoted as Singleton
codes.Comment: Accepted for publication in IJPA
A Method to determine Partial Weight Enumerator for Linear Block Codes
In this paper we present a fast and efficient method to find partial weight
enumerator (PWE) for binary linear block codes by using the error impulse
technique and Monte Carlo method. This PWE can be used to compute an upper
bound of the error probability for the soft decision maximum likelihood decoder
(MLD). As application of this method we give partial weight enumerators and
analytical performances of the BCH(130,66), BCH(103,47) and BCH(111,55)
shortened codes; the first code is obtained by shortening the binary primitive
BCH (255,191,17) code and the two other codes are obtained by shortening the
binary primitive BCH(127,71,19) code. The weight distributions of these three
codes are unknown at our knowledge.Comment: Computer Engineering and Intelligent Systems Vol 3, No.11, 201
Constacyclic Codes over Finite Fields
An equivalence relation called isometry is introduced to classify
constacyclic codes over a finite field; the polynomial generators of
constacyclic codes of length are characterized, where is the
characteristic of the finite field and is a prime different from
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