On the number of non-G-equivalent minimal abelian codes

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebrasF2k[A×Z2×Z2s]with respect to both the Euclidean and Hermitian inner products, wherekandsare positive integers andAis an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length2sover some Galois extensions of the ringF2k+uF2k, whereu2=0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of lengthpsoverFpk+uFpkare given. Combining these results, the complete enumeration of self-dual abelian codes inF2k[A×Z2×Z2s]is therefore obtained.

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Transform domain characterization of abelian codes

IEEE Transactions on Information Theory ◽

10.1109/18.165458 ◽

1992 ◽

Vol 38 (6) ◽

pp. 1817-1821 ◽

Cited By ~ 22

Author(s):

B.S. Rajan ◽

M.U. Siddiqi

Keyword(s):

Transform Domain ◽

Abelian Codes

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EXPLICIT DETERMINATION OF CERTAIN MINIMAL ABELIAN CODES AND THEIR MINIMUM DISTANCES

Asian-European Journal of Mathematics ◽

10.1142/s1793557112500027 ◽

2012 ◽

Vol 05 (01) ◽

pp. 1250002

Author(s):

Pooja Grover ◽

Ashwani K. Bhandari

Keyword(s):

Abelian Groups ◽

Cyclic Codes ◽

Group Algebras ◽

Primitive Idempotents ◽

Abelian Codes ◽

Complete Set ◽

Minimum Distances ◽

Minimal Codes ◽

Explicit Determination

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.

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