Nayene Michele Paião Panek

arXiv2021

Optimal Anticodes, Diameter Perfect Codes, Chains and Weights

Let $P$ be a partial order on $[n] = \{1,2,\ldots,n\}$, $\mathbb{F}_{q}^n$ be the linear space of $n$-tuples over a finite field $\mathbb{F}_{q}$ and $w$ be a weight on $\mathbb{F}_{q}$. In this paper, we consider metrics on $\mathbb{F}_{q}^n$ induced by chain orders $P$ over $[n]$ and weights $w$ over $\mathbb{F}_q$, and we determine the cardinality of all optimal anticodes and completely classify them. Moreover, we determine all diameter perfect codes for a set of relevant instances on the aforementioned metric spaces.

cs.IT↗

arXiv2017

Symmetry Group of Ordered Hamming Block Space

Let $P = (\{1,2,\ldots,n,\leq)$ be a poset that is an union of disjoint chains of the same length and $V=\mathbb{F}_q^N$ be the space of $N$-tuples over the finite field $\mathbb{F}_q$. Let $V_i = \mathbb{F}_q^{k_i}$, $1 \leq i \leq n$, be a family of finite-dimensional linear spaces such that $k_1+k_2+\ldots +k_n = N$ and let $V = V_1 \oplus V_2 \oplus \ldots \oplus V_n$ endow with the poset block metric $d_{(P,π)}$ induced by the poset $P$ and the partition $π=(k_1,k_2,\ldots,k_n)$, encompassing both Niederreiter-Rosenbloom-Tsfasman metric and error-block metric. In this paper, we give a complete description of group of symmetries of the metric space $(V,d_{(P,π)})$, called the ordered Hammming block space. In particular, we reobtain the group of symmetries of the Niederreiter-Rosenbloom-Tsfasman space and obtain the group of symmetries of the error-block metric space.

cs.IT↗