# Essential idempotents and simplex codes

Keywords:
Group code, Essential idempotent, Simplex code

### 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.

### References

S. D. Berman, Semisimple cyclic and abelian codes. II, Kibernetika 3(3) (1967) 21–30.

S. D. Berman, On the theory of group codes, Kibernetika 3(1) (1967) 31–39.

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984) 181–186.

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.

R. A. Ferraz, C. P. Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields Appl. 13(2) (2007) 382–393.

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.

J. Jensen, The concatenated structure of cyclic and abelian codes, IEEE Trans. Inform. Theory 31(6) (1985) 788–793.

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.

R. L. Miller, Minimal codes in abelian group algebras, J. Combinatorial Theory Ser A 26(2) (1979) 166–178.

C. P. Milies, F. D. de Melo, On cyclic and abelian codes, IEEE Trans. Information Theory 59(11) (2013) 7314–7319.

C. Polcino Milies, S. K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer Academic Publishers, Dortrecht, 2002.

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.

R. E. Sabin, On minimum distance bounds for abelian codes, Appl. Algebra Engrg. Comm. Comput. 3(3) (1992) 183–197.

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.

R. E. Sabin, S. J. Lomonaco, Metacyclic error–correcting codes, Appl. Algebra Engrg. Comm. Comput. 6(3) (1995) 191–210.

S. D. Berman, On the theory of group codes, Kibernetika 3(1) (1967) 31–39.

A. Bonisoli, Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin. 18 (1984) 181–186.

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.

R. A. Ferraz, C. P. Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields Appl. 13(2) (2007) 382–393.

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.

J. Jensen, The concatenated structure of cyclic and abelian codes, IEEE Trans. Inform. Theory 31(6) (1985) 788–793.

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.

R. L. Miller, Minimal codes in abelian group algebras, J. Combinatorial Theory Ser A 26(2) (1979) 166–178.

C. P. Milies, F. D. de Melo, On cyclic and abelian codes, IEEE Trans. Information Theory 59(11) (2013) 7314–7319.

C. Polcino Milies, S. K. Sehgal, An Introduction to Group Rings, Algebras and Applications, Kluwer Academic Publishers, Dortrecht, 2002.

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.

R. E. Sabin, On minimum distance bounds for abelian codes, Appl. Algebra Engrg. Comm. Comput. 3(3) (1992) 183–197.

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.

R. E. Sabin, S. J. Lomonaco, Metacyclic error–correcting codes, Appl. Algebra Engrg. Comm. Comput. 6(3) (1995) 191–210.

Published

2017-05-15

Section

Articles