Related papers: Two weight $\mathbb{Z}_{p^k}$-codes, $p$ odd prime

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight…

Linear codes have been an interesting subject of study for many years, as linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a class of…

We derive a general lower bound for the generalized Hamming weights of nested matrix-product codes, with a particular emphasis on the cases with two and three constituent codes. We also provide an upper bound which is reminiscent of the…

Let $\sigma(n)$ and $\gamma(n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\neq 1, 1782$ and $\sigma(n)=(\gamma(n))^2$, then there exist odd (not necessarily distinct)…

We obtain all possible parameters of Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4,$ together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions…

In this paper, we employ general results on the value distributions of perfect nonlinear functions from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$ together with a specific group action to give a unified approach to determining the weight…

Let m, k be positive integers, p be an odd prime and $\pi $ be a primitive element of $\mathbb{F}_{p^m}$. In this paper, we determine the weight distribution of a family of cyclic codes $\mathcal{C}_t$ over $\mathbb{F}_p$, whose duals have…

We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a $\{13,6,1;1,6,9\}$-CR code in $H(n,2)$, $n\ge 13$. We also…

The notions of strong, weak and dc-weak eigenforms mod $p^n$ was introduced and studied by Chen, Kiming and Wiese. In this work, we prove that there can be no uniform weight bound (that is, depending only on $p$, $n$) on dc-weak eigenforms…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

Let C be an additive subgroup of $\Z_{2k}^n$ for any $k\geq 1$. We define a Gray map $\Phi:\Z_{2k}^n \longrightarrow \Z_2^{kn}$ such that $\Phi(\codi)$ is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover,…

We prove that if two non-renormalizable cubic Siegel polynomials with bounded type rotation numbers are combinatorially equivalent, then they are also conformally equivalent. As a consequence, we show that in the one-parameter slice of…

If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of…

We provide an adelic relative trace formula proof to the Petersson/Bruggeman-Kuznetsov (PBK) formulas, specifically in the holomorphic case for $\kappa=2$ and the non-holomorphic case for $m_1m_2<0$. Given two sets of hypothesis on the non…

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their…

We proved recently (see \cite{lhgarasu}) the result on the title for odd prime divisors of such an $n.$ The result implies for many $n's$, more precisely, for an infinity of $n$'s with an arbitrary fixed number of prime divisors, the…

This note concerns the non-existence of three consecutive powerful numbers. We use Pell equations, elliptic curves, and second-order recurrences to show that there are no such triplets with the middle term a perfect cube and each of the…

In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$…

For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no…

Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.