35 research outputs found
Symplectic self-orthogonal quasi-cyclic codes
In this paper, we obtain sufficient and necessary conditions for quasi-cyclic
codes with index even to be symplectic self-orthogonal. Then, we propose a
method for constructing symplectic self-orthogonal quasi-cyclic codes, which
allows arbitrary polynomials that coprime to construct symplectic
self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic
codes, we provide lower and upper bounds on the minimum symplectic distances of
a class of 1-generator quasi-cyclic codes and their symplectic dual codes.
Finally, we construct many binary symplectic self-orthogonal codes with
excellent parameters, corresponding to 117 record-breaking quantum codes,
improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes.
http://www.codetables.de)
On the equivalence of linear cyclic and constacyclic codes
We introduce new sufficient conditions for permutation and monomial
equivalence of linear cyclic codes over various finite fields. We recall that
monomial equivalence and isometric equivalence are the same relation for linear
codes over finite fields. A necessary and sufficient condition for the monomial
equivalence of linear cyclic codes through a shift map on their defining set is
also given. Moreover, we provide new algebraic criteria for the monomial
equivalence of constacyclic codes over . Finally, we prove that
if , then all permutation equivalent constacyclic codes of
length over are given by the action of multipliers. The
results of this work allow us to prune the search algorithm for new linear
codes and discover record-breaking linear and quantum codes.Comment: 18 page
Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities
Let be the ring of integers modulo a prime number where
is a quadratic residue modulo . This paper presents the study of
constacyclic codes over chain rings and . We
also study additive constacyclic codes over and
using the generator polynomials over the
rings and respectively. Further, by defining Gray
maps on , and
we obtain some results on the Gray images of additive codes. Then we give the
weight enumeration and MacWilliams identities corresponding to the additive
codes over . Finally, as an application of
the obtained codes, we give quantum codes using the CSS construction.Comment: 22 page
Quasi-cyclic Hermitian construction of binary quantum codes
In this paper, we propose a sufficient condition for a family of 2-generator
self-orthogonal quasi-cyclic codes with respect to Hermitian inner product.
Supported in the Hermitian construction, we show algebraic constructions of
good quantum codes. 30 new binary quantum codes with good parameters improving
the best-known lower bounds on minimum distance in Grassl's code tables
\cite{Grassl:codetables} are constructed
New Quantum and LCD Codes over Finite Fields of Even Characteristic
For an integer m β₯ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m β image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .
 
Generating generalized necklaces and new quasi-cyclic codes
In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13)
Z_q(Z_q+uZ_q)-Linear Skew Constacyclic Codes
In this paper, we study skew constacyclic codes over the ring where , for a prime and We give the definition of these codes as subsets of the ring . Some structural properties of the skew polynomial ring are discussed, where is an automorphism of We describe the generator polynomials of skew constacyclic codes over also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over we obtained some new linear codes over . Finally, we have generalized these codes to double skew constacyclic codes over