Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userClassification of 8-divisible binary linear codes with minimum distance 24
137
0
0.0
(
0
)
Added by
Sascha Kurz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Sascha Kurz
Ask ChatGPT about the research
No Arabic abstract
We classify 8-divisible binary linear codes with minimum distance 24 and small length. As an application we consider the codes associated to nodal sextics with 65 ordinary double points.
rate research
Read More
The distance distribution of a code is the vector whose $i^text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $mathbb{F}_{q^n}^*$ on an $mathbb{F}_q$-subspace $U$ of $mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.
In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $Tr(x^d)$ over $F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $gcd(d, 2^{m}-1)=1$.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these codes support infinite families of 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m geqslant 5$ is odd and $3$-designs when $m geqslant 4$ is even, which provide infinite classes of simple $t$-designs with new parameters. Another significant class of $t$-designs we produce in this paper has supplementary designs with parameters 4-$(2^{2s+1}+ 1,5,5)$; these designs have the smallest index among all the known simple 4-$(q+1,5,lambda)$ designs derived from codes for prime powers $q$; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL$(2,2^{2s+1})$ as automorphism group constructed by Alltop in 1969.
We propose a binary message passing decoding algorithm for product codes based on generalized minimum distance decoding (GMDD) of the component codes, where the last stage of the GMDD makes a decision based on the Hamming distance metric. The proposed algorithm closes half of the gap between conventional iterative bounded distance decoding (iBDD) and turbo product decoding based on the Chase--Pyndiah algorithm, at the expense of some increase in complexity. Furthermore, the proposed algorithm entails only a limited increase in data flow compared to iBDD.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
