36 research outputs found
New quantum codes from self-dual codes over F_4
We present new constructions of binary quantum codes from quaternary linear
Hermitian self-dual codes. Our main ingredients for these constructions are
nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of
-dimensional binary quantum codes is provided. We give minimum distance
lower bounds for our quantum codes in terms of the minimum distance of their
ingredient linear codes. We also present new results on the minimum distance of
linear cyclic codes using their fixed subcodes. Finally, we list many new
record-breaking quantum codes obtained from our constructions.Comment: 16 page
Around Pelikan's conjecture on very odd sequences
Very odd sequences were introduced in 1973 by J. Pelikan who conjectured that
there were none of length >=5. This conjecture was disproved by MacWilliams and
Odlyzko in 1977 who proved there are in fact many very odd sequences. We give
connections of these sequences with duadic codes, cyclic difference sets,
levels (Stufen) of cyclotomic fields, and derive some new asymptotic results on
their lengths and on S(n), which denotes the number of very odd sequences of
length n.Comment: 21 pages, two tables. Revised version with improved presentation and
correction of some typos and minor errors that will appear in Manuscripta
Mathematic
A Generalization of the Tang-Ding Binary Cyclic Codes
Cyclic codes are an interesting family of linear codes since they have
efficient decoding algorithms and contain optimal codes as subfamilies.
Constructing infinite families of cyclic codes with good parameters is
important in both theory and practice. Recently, Tang and Ding [IEEE Trans.
Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] proposed an infinite family
of binary cyclic codes with good parameters. Shi et al. [arXiv:2309.12003v1,
2023] developed the binary Tang-Ding codes to the -ary case. Inspired by
these two works, we study -ary Tang-Ding codes, where . Good
lower bounds on the minimum distance of the -ary Tang-Ding codes are
presented. As a by-product, an infinite family of -ary duadic codes with a
square-root like lower bound is presented
Cyclotomic Constructions of Cyclic Codes with Length Being the Product of Two Primes
Cyclic codes are an interesting type of linear codes and have applications in
communication and storage systems due to their efficient encoding and decoding
algorithms. They have been studied for decades and a lot of progress has been
made. In this paper, three types of generalized cyclotomy of order two and
three classes of cyclic codes of length and dimension
are presented and analysed, where and are two distinct primes.
Bounds on their minimum odd-like weight are also proved. The three
constructions produce the best cyclic codes in certain cases.Comment: 19 page
Cyclic projective planes and binary, extended cyclic self-dual codes
AbstractIf P is a cyclic projective plane of order n, we give number theoretic conditions on n2 + n + 1 so that the binary code of P is contained in a binary cyclic code C whose extension is self-dual. When this containment occurs C does not contain any ovals of P. As a corollary to these conditions we obtain that the extended binary code of a cyclic projective plane of order 2s is contained in a binary extended cyclic self-dual code if and only if s is odd
Cyclic Codes from Cyclotomic Sequences of Order Four
Cyclic codes are an interesting subclass of linear codes and have been used
in consumer electronics, data transmission technologies, broadcast systems, and
computer applications due to their efficient encoding and decoding algorithms.
In this paper, three cyclotomic sequences of order four are employed to
construct a number of classes of cyclic codes over \gf(q) with prime length.
Under certain conditions lower bounds on the minimum weight are developed. Some
of the codes obtained are optimal or almost optimal. In general, the cyclic
codes constructed in this paper are very good. Some of the cyclic codes
obtained in this paper are closely related to almost difference sets and
difference sets. As a byproduct, the -rank of these (almost) difference sets
are computed