Essential idempotents and simplex codes

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 181 - 188, 10.01.2017

Gladys Chalom Raul A. Ferraz , Cesar Polcino Milies

https://doi.org/10.13069/jacodesmath.284931

Cited By: 3

Abstract

We define essential idempotents in group algebras and use them to prove that every mininmal abelian non-cyclic code is a repetition code. Also we use them to prove that every minimal abelian code is equivalent to a minimal cyclic code of the same length. Finally, we show that a binary cyclic code is simplex if and only if is of length of the form $n=2^k-1$ and is generated by an essential idempotent.

Keywords

Group code, Essential idempotent, Simplex code

References

• [1] S. D. Berman, Semisimple cyclic and abelian codes. II, Kibernetika 3(3) (1967) 21–30.
• [2] S. D. Berman, On the theory of group codes, Kibernetika 3(1) (1967) 31–39.
• [3] A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984) 181–186.
• [4] R. A. Ferraz, M. Guerreiro, C. P. Milies, G-equivalence in group algebras and minimal abelian codes, IEEE Trans. Inform. Theory 60(1) (2014) 252–260.
• [5] R. A. Ferraz, C. P. Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields Appl. 13(2) (2007) 382–393.
• [6] P. Grover, A. K. Bhandari, Explicit determination of certain minimal abelian codes and their minimum distance, Asian–European J. Math. 5(1) (2012) 1–24.
• [7] J. Jensen, The concatenated structure of cyclic and abelian codes, IEEE Trans. Inform. Theory 31(6) (1985) 788–793.
• [8] F. J. Mac Williams, Binary codes which are ideals in the group algebra of an abelian group, Bell System Tech. J. 49(6) (1970) 987–1011.
• [9] R. L. Miller, Minimal codes in abelian group algebras, J. Combinatorial Theory Ser A 26(2) (1979) 166–178.
• [10] C. P. Milies, F. D. de Melo, On cyclic and abelian codes, IEEE Trans. Information Theory 59(11) (2013) 7314–7319.
• [11] C. Polcino Milies, S. K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer Academic Publishers, Dortrecht, 2002.
• [12] A. Poli, Construction of primitive idempotents for a variable codes, Applied Algebra, Algorithmics and Error–Correcting Codes: 2nd International Conference, AAECC–2 Toulouse, France, October 1–5, 1984 Proceedings (1986) 25–35.
• [13] R. E. Sabin, On minimum distance bounds for abelian codes, Appl. Algebra Engrg. Comm. Comput. 3(3) (1992) 183–197.
• [14] R. E. Sabin, On determining all codes in semi–simple group rings, Applied Algebra, Algebraic Algorithms and Error–Correcting Codes: 10th International Symposium,AAECC-10 San Juan de Puerto Rico, Puerto Rico, May 10–14, 1993 Proceedings (1993) 279–290.
• [15] R. E. Sabin, S. J. Lomonaco, Metacyclic error–correcting codes, Appl. Algebra Engrg. Comm. Comput. 6(3) (1995) 191–210.