Non-abelian group

Non-abelian group

Elements of Galois Fields

Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012

Tolga M. Duman

Examples of groups include the set of integers Z under integer addition (which is an abelian group), and the set of all non-zero real numbers R\{0} under multiplication (also an abelian group). Nonsingular (square) matrices with real elements under matrix multiplication form a non-abelian group as matrix multiplication does not commute. On the other hand, the set of non-zero integers under multiplication or the set of non-negative integers under addition do not form a group (in both cases not all elements of the set have an inverse).

LDPC Codes Based on Rational Functions

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Journal Information

Published in IETE Journal of Research, 2021

Mohammad Gholami, Akram Nassaj

Define . Clearly, is a non-abelian group with the following product operation: Now, for a given exponent matrix E of a RLDPC code with the elements belong to , by an extension of E, we mean the exponent matrix derived by multiplying the rows or columns of E by some elements of . By Lemma 3.1, it can be seen easily that the RLDPC code with the exponent matrix E is equivalent with an APM-LDPC code if and only if there is an extension of E, say , such that all of the elements of belong to the set . In continue, type-I and type-II RLDPC codes are precedented which are and are not equivalent with APM-LDPC codes,respectively.

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Elements of Galois Fields

LDPC Codes Based on Rational Functions