141 research outputs found
A class of narrow-sense BCH codes over of length
BCH codes with efficient encoding and decoding algorithms have many
applications in communications, cryptography and combinatorics design. This
paper studies a class of linear codes of length over
with special trace representation, where is an odd prime
power. With the help of the inner distributions of some subsets of association
schemes from bilinear forms associated with quadratic forms, we determine the
weight enumerators of these codes. From determining some cyclotomic coset
leaders of cyclotomic cosets modulo , we prove
that narrow-sense BCH codes of length with designed distance
have the corresponding trace representation, and have the
minimal distance and the Bose distance , where
The Subfield Codes of Some Few-Weight Linear Codes
Subfield codes of linear codes over finite fields have recently received a
lot of attention, as some of these codes are optimal and have applications in
secrete sharing, authentication codes and association schemes. In this paper,
the -ary subfield codes of six different families of
linear codes are presented, respectively. The parameters and
weight distribution of the subfield codes and their punctured codes
are explicitly determined. The parameters of the duals of
these codes are also studied. Some of the resultant -ary codes
and their dual codes are optimal
and some have the best known parameters. The parameters and weight enumerators
of the first two families of linear codes are also settled,
among which the first family is an optimal two-weight linear code meeting the
Griesmer bound, and the dual codes of these two families are almost MDS codes.
As a byproduct of this paper, a family of quaternary
Hermitian self-dual code are obtained with . As an application,
several infinite families of 2-designs and 3-designs are also constructed with
three families of linear codes of this paper.Comment: arXiv admin note: text overlap with arXiv:1804.06003,
arXiv:2207.07262 by other author
Optimizing Linear Correctors: A Tight Output Min-Entropy Bound and Selection Technique
Post-processing of the raw bits produced by a true random number generator
(TRNG) is always necessary when the entropy per bit is insufficient for
security applications. In this paper, we derive a tight bound on the output
min-entropy of the algorithmic post-processing module based on linear codes,
known as linear correctors. Our bound is based on the codes' weight
distributions, and we prove that it holds even for the real-world noise sources
that produce independent but not identically distributed bits. Additionally, we
present a method for identifying the optimal linear corrector for a given input
min-entropy rate that maximizes the throughput of the post-processed bits while
simultaneously achieving the needed security level. Our findings show that for
an output min-entropy rate of , the extraction efficiency of the linear
correctors with the new bound can be up to higher when compared to
the old bound, with an average improvement of over the entire input
min-entropy range. On the other hand, the required min-entropy of the raw bits
for the individual correctors can be reduced by up to
The extended codes of a family of reversible MDS cyclic codes
A linear code with parameters is called a maximum distance
separable (MDS for short) code. A linear code with parameters is
said to be almost maximum distance separable (AMDS for short). A linear code is
said to be near maximum distance separable (NMDS for short) if both the code
and its dual are AMDS. MDS codes are very important in both theory and
practice. There is a classical construction of a MDS code
for each with , which is a
reversible and cyclic code. The objective of this paper is to study the
extended codes of this family of MDS codes. Two families of MDS codes and
several families of NMDS codes are obtained. The NMDS codes have applications
in finite geometry, cryptography and distributed and cloud data storage
systems. The weight distributions of some of the extended codes are determined
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