Metacyclic error-correcting codes

Sabin, Roberta Evans; Lomonaco, Samuel J.

doi:10.1007/bf01195337

Cited by 36 publications

(29 citation statements)

References 5 publications

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“…Using this we prove that if G = AB, for A and B two abelian subgroups of G then every G-code is also an abelian group code. This extends a result of Sabin and Lomonaco [4]. We present some examples showing that this cannot be extended to left group codes.…”

supporting

confidence: 87%

“…Moreover, i = g(i 0 ) for a unique g ∈ G. Therefore g Recall that a group G is metacyclic if it has a cyclic normal subgroup N such that G/N is cyclic. Sabin and Lomonaco [4] have proved that if C is a G-code of a split metacyclic group G (i.e. G is a semidirect product of cyclic groups) then C is an abelian group code.…”

Section: Group Codes Versus Abelian Group Codesmentioning

confidence: 99%

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An intrinsical description of group codes

Bernal

Río

Simón

2009

Des. Codes Cryptogr.

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Abstract. A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism FG → F n which maps G to the standard basis of F n . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space F n , which does not assume an "a priori" group algebra structure on F n . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code.It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

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supporting

confidence: 87%

Section: Group Codes Versus Abelian Group Codesmentioning

confidence: 99%

An intrinsical description of group codes

Bernal

Río

Simón

2009

Des. Codes Cryptogr.

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“…Ideals I 1 ⊂ FG 1 and I 2 ⊂ FG 2 such that γ(I 1 ) = I 2 are thus equivalent, in the sense that they have the same dimension and the same weight distribution. In this case, the codes I 1 and I 2 are said to be permutation equivalent and were called combinatorially equivalent in [15].…”

Section: The Equivalencementioning

confidence: 99%

“…Also, Sabin and Lomonaco [15] have shown that central codes in metacyclic group algebras are equivalent to abelian codes.…”

Section: Introductionmentioning

confidence: 99%

Essential idempotents and simplex codes

Chalom¹,

Ferraz²,

Milies³

2017

Journal of Algebra Combinatorics Discrete Structures and Applications

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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.

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“…The group algebra F q [G] and its Wedderburn decomposition are not only of interest in pure algebra, they also have applications in coding theory. Cyclic codes can be realized as ideals of group algebras over cyclic groups [8] and many other important codes appear as ideals of noncyclic group algebras [2,8,9]. With the concrete realization of the Wedderburn decomposition of F q [G], it is straightforward to produce all the ideals of F q [G].…”

Section: Introductionmentioning

confidence: 99%