Conference abstracts

Session S07 - Finite Fields

No date set

## Similarity between the algebraic structure associated with projective space and combinatorial design via Hasse diagram

### CPAq/UFMS - FEEC/UNICAMP, Brasil   -   leandro.lima@ufms.br

Combinatorial design is an important combinatorial structure having a high degree of regularity and which is related to the existence and construction of systems of sets with finite cardinality, [1]. As examples we mention the existing relationship between error-correcting codes in the Hamming space and combinatorial design, where the codewords of weight 3 of the Hamming code form a triple Steiner system STS(7), a projective plane of order 2, known as the Fano plane, [2], as well as $q$-analogs of a code whose codewords have constant Hamming weight in the Hamming space, a code belonging to a Grassmannian in the projective space, [3,4]. Projective space of order $m$ over a finite field $\mathbb{F}_p$, denoted by $\mathcal{P}(\mathbb{F}_{p}^{m})$, (note that $\mathbb{F}_{p^m}$ is isomorphic to $\mathbb{F}_{p}^{m}$), is the set of all the subspaces in the vector space $\mathbb{F}_{p}^{m}$. The projective space endowed with the subspace distance $d(X,Y)=dim(X)+dim(Y)-2dim(X \cap Y)$ is a metric space. Hence, the subspace code $\mathcal{C}$ with parameters $(n,M,d)$ in the projective space is a subset of $\mathcal{P}(\mathbb{F}_{p}^{m})$ with cardinality $M$ with a subspace distance at least $d$ between any two codewords, [5]. In this paper we show the existing similarity between the Hasse diagram of an Abelian group consisting of the product of multiplicative finite Abelian groups $\mathbb{Z}_p^m$ and the Hasse diagram of the projective space $\mathcal{P}(\mathbb{F}_{p}^{m})$, with the aim to provide the elements that may be useful in the identification and in the construction of good subspaces codes, [6].

[1] - D.R. Stinson, \textit{Combinatorial Designs: Constructions and Analysis}, Springer Verlag, New York, USA, 2004.

[2]-T.Etzion and N. Silberstein, "Error-Correcting codes in projective spaces via rank metric codes and Ferrers diagrams," \textit{ IEEE Trans. Inform. Theory}, vol. 55, n.º7, pp.2909-2919, Jul. 2009.

[3]-M.Braun, T. Etzion, P.R.J. Ostergard, A. Vardy, and A. Warssermann, "Existence of q-analogs of Steiner sstems," arxiv.org/abs/1304.1462, Apr. 2013.

[4]-T. Etzion and A. Vardy, "Error-Correcting codes in projective space," \textit{IEEE Intl. Symp. on Inform. Theory - ISIT-08}, pp. 871-875, Toronto, Canada, Jul. 2008.

[5]-A. Khaleghi, D. Silva, and F.R. Kschischang, "Subspace codes," \textit{Lecture Notes in Computer Science}, vol. 5921, pp. 1-21, 2009.

[6]-C.H.A. Costa e M. Guerreiro, "Automorphisms of finite Abelian groups,", MS thesis, Mathematics Dept, UFV, Vi\c{c}osa, Minas Gerais, 2014. (in Portuguese)

Joint work with Reginaldo Palazzo Jr. (FEEC/UNICAMP) e-mail: palazzo@dt.fee.unicamp.br.