Araştırma Makalesi
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Essential idempotents and simplex codes

Yıl 2017, , 181 - 188, 10.01.2017
https://doi.org/10.13069/jacodesmath.284931

Öz

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.

Kaynakça

  • [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.
Yıl 2017, , 181 - 188, 10.01.2017
https://doi.org/10.13069/jacodesmath.284931

Öz

Kaynakça

  • [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.
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Konular Mühendislik
Bölüm Makaleler
Yazarlar

Gladys Chalom Bu kişi benim

Raul A. Ferraz

Cesar Polcino Milies

Yayımlanma Tarihi 10 Ocak 2017
Yayımlandığı Sayı Yıl 2017

Kaynak Göster

APA Chalom, G., Ferraz, R. A., & Milies, C. P. (2017). Essential idempotents and simplex codes. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 181-188. https://doi.org/10.13069/jacodesmath.284931
AMA Chalom G, Ferraz RA, Milies CP. Essential idempotents and simplex codes. Journal of Algebra Combinatorics Discrete Structures and Applications. Mayıs 2017;4(2 (Special Issue: Noncommutative rings and their applications):181-188. doi:10.13069/jacodesmath.284931
Chicago Chalom, Gladys, Raul A. Ferraz, ve Cesar Polcino Milies. “Essential Idempotents and Simplex Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, sy. 2 (Special Issue: Noncommutative rings and their applications) (Mayıs 2017): 181-88. https://doi.org/10.13069/jacodesmath.284931.
EndNote Chalom G, Ferraz RA, Milies CP (01 Mayıs 2017) Essential idempotents and simplex codes. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 181–188.
IEEE G. Chalom, R. A. Ferraz, ve C. P. Milies, “Essential idempotents and simplex codes”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 2 (Special Issue: Noncommutative rings and their applications), ss. 181–188, 2017, doi: 10.13069/jacodesmath.284931.
ISNAD Chalom, Gladys vd. “Essential Idempotents and Simplex Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (Mayıs 2017), 181-188. https://doi.org/10.13069/jacodesmath.284931.
JAMA Chalom G, Ferraz RA, Milies CP. Essential idempotents and simplex codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:181–188.
MLA Chalom, Gladys vd. “Essential Idempotents and Simplex Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 4, sy. 2 (Special Issue: Noncommutative rings and their applications), 2017, ss. 181-8, doi:10.13069/jacodesmath.284931.
Vancouver Chalom G, Ferraz RA, Milies CP. Essential idempotents and simplex codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):181-8.