arXiv subjects

Josu Sangroniz

Publications and source records attributed to Josu Sangroniz.

2 recordsLinked to original sources

arXiv2019

The GC-content of a family of cyclic codes with applications to DNA-codes

Given a prime power $q$ and a positive integer $r>1$ we say that a cyclic code of length $n$, $C\subseteq F_{q^r}^n$, is Galois supplemented if for any non-trivial element $σ$ in the Galois group of the extension $ F_{q^r}/ F_q$, $C+C^σ= F_{q^r}^n$, where $C^σ=\{(x_1^σ,\dots,x_n^σ)\mid (x_1,\dots,x_n)\in C\}$. This family includes the quadratic-residue (QR) codes over $ F_{q^2}$. Some important properties QR-codes are then extended to Galois supplemented codes and a new one is also considered, which is actually the motivation for the introduction of this family of codes: in a Galois supplemented code we can explicitly count the number of words that have a fixed number of coordinates in $ F_q$. In connection with DNA-codes the number of coordinates of a word in $ F_4^n$ that lie in $ F_2$ is sometimes referred to as the $GC$-content of the word and codes over $ F_4$ all of whose words have the same $GC$-content have a particular interest. Therefore our results have some direct applications in this direction.

cs.IT↗

arXiv2016

Words and characters in finite p-groups

Given a group word $w$ in $k$ variables, a finite group $G$ and $g\in G$, we consider the number $N_{w,G}(g)$ of $k$-tuples $g_1,\dots ,g_k$ of elements of $G$ such that $w(g_1,\dots ,g_k)=g$. In this work we study the functions $N_{w,G}$ for the class of nilpotent groups of nilpotency class $2$. We show that, for the groups in this class, $N_{w,G}(1)\geq |G|^{k-1}$, an inequality that can be improved to $N_{w,G}(1)\geq |G|^k/|G_w|$ ($G_w$ is the set of values taken by $w$ on $G$) if $G$ has odd order. This last result is explained by the fact that the functions $N_{w,G}$ are characters of $G$ in this case. For groups of even order, all that can be said is that $N_{w,G}$ is a generalized character, something that is false in general for groups of nilpotency class greater than $2$. We characterize group theoretically when $N_{x^n,G}$ is a character if $G$ is a $2$-group of nilpotency class $2$. Finally we also address the (much harder) problem of studying if $N_{w,G}(g)\geq |G|^{ k-1}$ for $g\in G_w$, proving that this is the case for the free $p$-groups of nilpotency class $2$ and exponent $p$.

math.GR↗