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In this paper, we study no 4-cycle, high-rate LDPC codes based on finite geometries for use in data storage devices and prove that these codes cannot be classified as quasi-cyclic (QC) codes but should be considered as broader generalized quasi-cyclic (GQC) codes. Because of the GQC structure of such codes, they can be systematically encoded using Groebner bases and their encoder can be implemented...
In this paper, a fundamental lemma in algebraic coding theory is established, which is frequently appeared in the encoding and decoding for algebraic codes such as Reed-Solomon and algebraic geometry codes. This lemma states that two vector spaces, one corresponds to information symbols and the other is indexed by the support of Gröbner basis, are canonically isomorphic, and moreover, the isomorphism...
In this study, we proved that several promising classes of codes based on finite geometries cannot be classified as quasi-cyclic (QC) codes but should be included in broader generalized quasi-cyclic (GQC) codes. Further, we proposed an algorithm (transpose algorithm) for the computation of the Grobner bases from the parity check matrices of GQC codes. Because of the GQC structure of such codes, they...
We define generalized quasi-cyclic (GQC) codes as linear codes with nontrivial automorphism groups. Therefore, GQC codes, unlike quasi-cyclic codes, can include many important codes such as Hermitian and projective geometry (PG) codes; this capability is important in practical applications. Further, we propose the echelon canonical form algorithm for computing Grobner bases from their parity check...
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