Group and Loop Algebras over Finite Fields

Raul Antonio Ferraz

ISBN 10: 3639512685 / ISBN 13: 9783639512687

Editore: SPS, 2015

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nach der Bestellung gedruckt Neuware - Printed after ordering - In this, the 15th year of the 21st century, the need to translate plain text into code and to transmit securely, and the plethora of uses to which encryption is applied, needs no amplification. Recent methods have produced so-called cyclic codes, which have the property that if (a_1,a_2,a_3,.,a_n) is a code word, then so is the cyclic shift (a_n,a_1,a_2,.,a_{n-1}). The realization that such a code is an ideal in the group algebra of a cyclic group has motivated a large body of research into using group algebras to produce codes. In this area, the central theme is ``efficiency,'' a term that describes the need to maximize simultaneously both the Hamming distance between code words and the dimension of the code. Since an ideal in a semisimple group algebra KG corresponds to a central idempotent, and since a central idempotent is the sum of idempotents that generate the simple components of KG, the need to find these simple components is clear. This is the theme of the present monograph, which will interest many algebraists, including those who study Moufang loops and alternative rings. The groups studied are the building blocks of loops whose loop rings are alternative.

Codice articolo 9783639512687

Dati bibliografici

• Titolo: Group and Loop Algebras over Finite Fields
• Editore: SPS
• Data di pubblicazione: 2015
• Lingua: Inglese
• ISBN 10: 3639512685
• ISBN 13: 9783639512687
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 220 L x 150 A x 10 P (millimetri) 220 millimetri di larghezza per 150 millimetri di altezza per 10 millimetri di profondità
• Peso: 238 grammi

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Riassunto

In this, the 15th year of the 21st century, the need to translate plain text into code and to transmit securely, and the plethora of uses to which encryption is applied, needs no amplification. Recent methods have produced so-called cyclic codes, which have the property that if (a_1,a_2,a_3,...,a_n) is a code word, then so is the cyclic shift (a_n,a_1,a_2,...,a_{n-1}). The realization that such a code is an ideal in the group algebra of a cyclic group has motivated a large body of research into using group algebras to produce codes. In this area, the central theme is ``efficiency,'' a term that describes the need to maximize simultaneously both the Hamming distance between code words and the dimension of the code. Since an ideal in a semisimple group algebra KG corresponds to a central idempotent, and since a central idempotent is the sum of idempotents that generate the simple components of KG, the need to find these simple components is clear. This is the theme of the present monograph, which will interest many algebraists, including those who study Moufang loops and alternative rings. The groups studied are the building blocks of loops whose loop rings are alternative.