113 research outputs found
Quasi-Cyclic Complementary Dual Code
LCD codes are linear codes that intersect with their dual trivially. Quasi
cyclic codes that are LCD are characterized and studied by using their
concatenated structure. Some asymptotic results are derived. Hermitian LCD
codes are introduced to that end and their cyclic subclass is characterized.
Constructions of QCCD codes from codes over larger alphabets are given
Quasi-cyclic subcodes of cyclic codes
We completely characterize possible indices of quasi-cyclic subcodes in a
cyclic code for a very broad class of cyclic codes. We present enumeration
results for quasi-cyclic subcodes of a fixed index and show that the problem of
enumeration is equivalent to enumeration of certain vector subspaces in finite
fields. In particular, we present enumeration results for quasi-cyclic subcodes
of the simplex code and duals of certain BCH codes. Our results are based on
the trace representation of cyclic codes
Quasi-Cyclic Codes
Quasi-cyclic codes form an important class of algebraic codes that includes
cyclic codes as a special subclass. This chapter focuses on the algebraic
structure of quasi-cyclic codes, first. Based on these structural properties,
some asymptotic results, a few minimum distance bounds and further applications
such as the trace representation and characterization of certain subfamilies of
quasi-cyclic codes are elaborated. This survey will appear as a chapter in "A
Concise Encyclopedia of Coding Theory" to be published by CRC Press.Comment: arXiv admin note: text overlap with arXiv:1906.0496
Linear Complementary Pair Of Group Codes over Finite Chain Rings
Linear complementary dual (LCD) codes and linear complementary pair (LCP) of
codes over finite fields have been intensively studied recently due to their
applications in cryptography, in the context of side-channel and fault
injection attacks. The security parameter for an LCP of codes is
defined as the minimum of the minimum distances and . It has
been recently shown that if and are both 2-sided group codes over a
finite field, then and are permutation equivalent. Hence the
security parameter for an LCP of 2-sided group codes is simply .
We extend this result to 2-sided group codes over finite chain rings
Further improvements on the designed minimum distance of algebraic geometry codes
In the literature about algebraic geometry codes one finds a lot of results improving Goppa’s minimum distance bound. These improvements often use the idea of “shrinking” or “growing” the defining divisors of the codes under certain technical conditions. The main contribution of this article is to show that most of these improvements can be obtained in a unified way from one (rather simple) theorem. Our result does not only simplify previous results but it also improves them further
Artin-Schreier curves and weights of two dimensional cyclic codes
Let GF(q) be the finite field with q elements of characteristic p, GF(q^m) be the extension of degree m>1 and f(x) be a polynomial over GF(q^m). We determine a necessary and sufficient condition for y^q-y=f(x) to have the maximum number of affine GF(qm)-rational points. Then we study the weights of 2-D cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin-Schreier curves.We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds
A Bound on the number of rational points of certain Artin-Schreier families
Let F[sub q] = F[sub p]l for some 1 > 0 and consider the extension F[sub q][sup m] with m > 1. We consider families of curves of the form F = {y[sup q] - y = λ[sub 1]x[sup i1]+ λ[sub 2]x[sup i2] + ··· + λ[sub s]x[sup is]; λ[sub j] ∈ F[sub q]m, i[sub j] > 0 }. We call such families Artin-Schreier families, even though not every curve in F need be an Artin-Schreier curve. It is easy to see that the members of such a family can have at most q[sup m+l] affine F[sub q[sup m]]-rational points. Using a well-known coding theory technique, we determine the condition under which F can attain this bound and we obtain some simple, but interesting, corollaries of this result. One of these consequences shows the existence of maximal curves of Artin-Schreier type. Our main result is important for minimum distance analysis of certain two-dimensional cyclic code
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