Essential idempotents and simplex codes

Abstract

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.

Group algebras and coding theory: a short survey

“…In this very short survey we cover only one aspect of this role: we focus on the relationship between weight and dimension of group codes. This type of codes have recently been the object of active research (see [2], [6], [7], [8], [9], [10], [11], [12], [13], [19], [20], [24], [27]).…”

Essential idempotents and codes of constant weight

“…Recall that a binary linear code of dimension k and length n is called simplex if a generating matrix for the code contains all possible non zero columns of length k. Since these are 2 k − 1 in number, this matrix must be of size k × (2 k − 1) so, we must have n = 2 k − 1. It was also shown in [5,Theorem 4.4] that a binary linear code of dimension k and length n = 2 k − 1 is a simplex code if and only if it is essencial. As a consequence, and taking the results in [4] into account, it follows that every binary linear code of constant weight is a repetition of a code generated by an essential idempotent.…”

“…This type of idempotents was introduced in [5] where it was shown that, if e is not essential, then the code FAe is a repetition code and also that a primitive idempotent e is essential if and only if the map π : G → Ge is a group isomorphism. As a consequence, it follows that if A is abelian and FA contains an essential idempotent, then A is cyclic [5,Corollary 2.5].…”

“…This type of idempotents was introduced in [5] where it was shown that, if e is not essential, then the code FAe is a repetition code and also that a primitive idempotent e is essential if and only if the map π : G → Ge is a group isomorphism. As a consequence, it follows that if A is abelian and FA contains an essential idempotent, then A is cyclic [5,Corollary 2.5]. It can be shown that every minimal ideal in the semi-simple group algebra FA of a finite abelian group A is equivalent to a minimal ideal in the group algebra FC of a cyclic group C of the same order [5,Theorem 3.3].…”

“…As a consequence, it follows that if A is abelian and FA contains an essential idempotent, then A is cyclic [5,Corollary 2.5]. It can be shown that every minimal ideal in the semi-simple group algebra FA of a finite abelian group A is equivalent to a minimal ideal in the group algebra FC of a cyclic group C of the same order [5,Theorem 3.3].…”