# American Institute of Mathematical Sciences

November  2020, 14(4): 631-650. doi: 10.3934/amc.2020035

## Abelian non-cyclic orbit codes and multishot subspace codes

 1 Department of Mathematics and Statistics, Federal University of São João del-Rei, Praça Frei Orlando, 170, Centro, São João del-Rei - MG, 36307-352, Brazil 2 Department of Communications, FEEC, State University of Campinas, Cidade Universitária Zeferino Vaz - Barão Geraldo, Campinas - SP, 13083-852, Brazil 3 Department of Mathematics, Federal University of Viçosa, Avenida Peter Henry Rolfs, s/n, Viçosa - MG, 36570-900, Brazil

Received  November 2018 Revised  June 2019 Published  November 2019

Fund Project: The first author was supported by CAPES and CNPq PhD scholarships

In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, Abelian orbit codes are defined and a construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established.

An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.

Citation: Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035
##### References:

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##### References:
Interdistance sets $D\left( {\left\{ V \right\},{C_H}\left( {{\alpha ^i}V} \right)} \right)$, for $1 \leq i \leq 4$
 $d_S (.,.)$ $\alpha^1$ $\alpha^{10}$ $\alpha^{19}$ $\alpha^{28}$ $\alpha^{37}$ $\alpha^{46}$ $\alpha^{55}$ $\alpha^0$ 4 4 6 6 6 4 6 $d_S (.,.)$ $\alpha^2$ $\alpha^{11}$ $\alpha^{20}$ $\alpha^{29}$ $\alpha^{38}$ $\alpha^{47}$ $\alpha^{56}$ $\alpha^0$ 4 6 4 4 6 4 6 $d_S (.,.)$ $\alpha^3$ $\alpha^{12}$ $\alpha^{21}$ $\alpha^{30}$ $\alpha^{39}$ $\alpha^{48}$ $\alpha^{57}$ $\alpha^0$ 4 4 6 4 4 4 4 $d_S (.,.)$ $\alpha^4$ $\alpha^{13}$ $\alpha^{22}$ $\alpha^{31}$ $\alpha^{40}$ $\alpha^{49}$ $\alpha^{58}$ $\alpha^0$ 4 6 6 4 4 6 4
 $d_S (.,.)$ $\alpha^1$ $\alpha^{10}$ $\alpha^{19}$ $\alpha^{28}$ $\alpha^{37}$ $\alpha^{46}$ $\alpha^{55}$ $\alpha^0$ 4 4 6 6 6 4 6 $d_S (.,.)$ $\alpha^2$ $\alpha^{11}$ $\alpha^{20}$ $\alpha^{29}$ $\alpha^{38}$ $\alpha^{47}$ $\alpha^{56}$ $\alpha^0$ 4 6 4 4 6 4 6 $d_S (.,.)$ $\alpha^3$ $\alpha^{12}$ $\alpha^{21}$ $\alpha^{30}$ $\alpha^{39}$ $\alpha^{48}$ $\alpha^{57}$ $\alpha^0$ 4 4 6 4 4 4 4 $d_S (.,.)$ $\alpha^4$ $\alpha^{13}$ $\alpha^{22}$ $\alpha^{31}$ $\alpha^{40}$ $\alpha^{49}$ $\alpha^{58}$ $\alpha^0$ 4 6 6 4 4 6 4
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