Related papers: On the shape of the general error locator polynomi…
Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This…
Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems…
The cyclic code is a subclass of linear codes and has applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open…
The purpose of this paper is to study the cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$. After providing the generator polynomial of cyclic self orthogonal codes over $\mathbb{Z}_{p^m}$, we give the necessary and sufficient condition…
The square $C^{*2}$ of a linear error correcting code $C$ is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in $C$. Squares of codes have gained attention for several applications…
Let $r,s,t$ be three positive integers and $\mathcal{C}$ be a binary linear code of lenght $r+s+t$. We say that $\mathcal{C}$ is a triple cyclic code of lenght $(r,s,t)$ over $\mathbb{Z}_2$ if the set of coordinates can be partitioned into…
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.…
We consider the problem of constructing a cyclic listing of all bitstrings of length $2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging…
A lot of attention has been paid to the investigation of the algebraic properties of linear codes. In most cases, this investigation involves the determination of required code automorphisms, which are useful for decoders, such as the…
We present algorithms performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Based on the initial work by Comer, Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial…
Due to their efficient encoding and decoding algorithms cyclic codes, a subclass of linear codes, have applications in consumer electronics, data storage systems, and communication systems. In this paper, Dickson polynomials of the first…
In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic…
We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear…
We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + \gamma)^{\ell}$ and $(x^n - \xi)^i(x^n + \xi)^j$\ respectively, where $x^n + \gamma$, $x^n - \xi$ and $x^n + \xi$ are irreducible over…
Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their…
Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open…
This note is a stripped down version of a published paper on the Potts partition function, where we concentrate solely on the linear coding aspect of our approach. It is meant as a resource for people interested in coding theory but who do…
In this paper, cyclic codes are established over some finite quaternion integer rings with respect to the quaternion Mannheim distance, and de- coding algorithm for these codes is given.
It is well known that the problem of determining the weight distributions of families of cyclic codes is, in general, notoriously difficult. An even harder problem is to find characterizations of families of cyclic codes in terms of their…
A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special…