Geodesic
Basic Reed--Muller codes as group codes
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 137-154.

Voir la notice de l'article provenant de la source Math-Net.Ru

Reed–Muller codes are one of the most well-studied families of codes, however, there are still open problems regarding their structure. Recently, a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. We recall that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. In this paper, we prove that basic Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers. This implies the same result for regular codes. Also we show how to describe the inclusion graph of basic Reed–Muller codes and radical powers via simple arithmetic equations. 
@article{FPM_2013_18_4_a10,
     author = {I. N. Tumaykin},
     title = {Basic {Reed--Muller} codes as group codes},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {137--154},
     publisher = {mathdoc},
     volume = {18},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a10/}
}
TY  - JOUR
AU  - I. N. Tumaykin
TI  - Basic Reed--Muller codes as group codes
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2013
SP  - 137
EP  - 154
VL  - 18
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a10/
LA  - ru
ID  - FPM_2013_18_4_a10
ER  - 
%0 Journal Article
%A I. N. Tumaykin
%T Basic Reed--Muller codes as group codes
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2013
%P 137-154
%V 18
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a10/
%G ru
%F FPM_2013_18_4_a10
I. N. Tumaykin. Basic Reed--Muller codes as group codes. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 137-154. http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a10/

[1] Assmus E. F. (Jr.), Key J. D., “Polynomial codes and finite geometries”, Handbook of Coding Theory, v. 2, Elsevier, Amsterdam, 1998, 1269–1343 | MR | Zbl

[2] Berman S. D., “On the theory of group codes”, Kibernetika, 3, 1967, 31–39 | MR

[3] Couselo E., Gonzalez S., Markov V., Martinez C., Nechaev A., “Ideal representation of Reed–Solomon and Reed–Muller codes”, Algebra Logic, 51:3 (2012), 195–212 | DOI | MR | Zbl

[4] Jennings S. A., “The structure of the group ring of a $p$-group over a modular field”, Trans. Am. Math. Soc., 50 (1941), 175–185 | MR | Zbl

[5] Landrock P., Manz O., “Classical codes as ideals in group algebras”, Designs, Codes Cryptography, 2:3 (1992), 273–285 | DOI | MR | Zbl