133 research outputs found
New binary and ternary LCD codes
LCD codes are linear codes with important cryptographic applications.
Recently, a method has been presented to transform any linear code into an LCD
code with the same parameters when it is supported on a finite field with
cardinality larger than 3. Hence, the study of LCD codes is mainly open for
binary and ternary fields. Subfield-subcodes of -affine variety codes are a
generalization of BCH codes which have been successfully used for constructing
good quantum codes. We describe binary and ternary LCD codes constructed as
subfield-subcodes of -affine variety codes and provide some new and good LCD
codes coming from this construction
On the weight distributions of several classes of cyclic codes from APN monomials
Let be an odd integer and be an odd prime. % with ,
where is an odd integer.
In this paper, many classes of three-weight cyclic codes over
are presented via an examination of the condition for the
cyclic codes and , which have
parity-check polynomials and respectively, to
have the same weight distribution, where is the minimal polynomial of
over for a primitive element of
. %For , the duals of five classes of the proposed
cyclic codes are optimal in the sense that they meet certain bounds on linear
codes. Furthermore, for and positive integers such
that there exist integers with and satisfying , the value
distributions of the two exponential sums T(a,b)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in
\mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where , are
settled. As an application, the value distribution of is utilized to
investigate the weight distribution of the cyclic codes
with parity-check polynomial . In the case of and
even satisfying the above condition, the duals of the cyclic codes
have the optimal minimum distance
Five Families of Three-Weight Ternary Cyclic Codes and Their Duals
As a subclass of linear codes, cyclic codes have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, five families of
three-weight ternary cyclic codes whose duals have two zeros are presented. The
weight distributions of the five families of cyclic codes are settled. The
duals of two families of the cyclic codes are optimal
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
- …