Polynomial codes

Kasami, Tadao; Lin, Shu; Peterson, W. W.

doi:10.1109/tit.1968.1054226

Cited by 104 publications

(38 citation statements)

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“…We call this code the type-I -dimensional EG-LDPC code. Since the column weight of is the minimum distance of is at least It turns out that this EG-LDPC code is the one-step majoritylogic decodable th-order EG code constructed based on EG [28], [39], [40] and is the dual code of a polynomial code [40]- [43]. Therefore, it is cyclic and its generator polynomial is completely characterized by its roots in GF .…”

Section: A Type-i Eg-ldpc Codesmentioning

confidence: 99%

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Kou

Lin

Fossorier

2001

IEEE Trans. Inform. Theory

1,112

824

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Abstract-This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.

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Section: A Type-i Eg-ldpc Codesmentioning

confidence: 99%

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Kou

Lin

Fossorier

2001

IEEE Trans. Inform. Theory

1,112

824

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“…Let a be a primitive element in F p ms . Then, the roots of generator polynomial of C ð1Þ EG;c ðm; m; 0; s; pÞ are given by Kasami & Lin (1971, theorem 6), see also Kasami et al (1968b) and Lin & Costello (2004),…”

Section: ) (mentioning

confidence: 99%

Asymmetric quantum codes: constructions, bounds and performance

Sarvepalli

Klappenecker

Rötteler³

2009

Proc. R. Soc. A.

125

147

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Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit-and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank-Shor-Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose-ChaudhuriHocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

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“…In next the factorization of Polynomials over Galois Field GF(p q ) had been elaborated [12]. Later Appropriate Coding Techniques of Polynomials over Galois Field GF(p q ) had been illustrated with example [13]. The previous idea of factorizing Polynomials over Galois Field GF(p q ) [12] had also been extended to Large value of P or Large Finite fields [14].…”

Section: Literature Surveymentioning

confidence: 99%

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

Dey¹,

Ghosh²

2018

OJDM

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Polynomial codes

Cited by 104 publications

References 10 publications

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Asymmetric quantum codes: constructions, bounds and performance

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field <I>GF</I>(<I>p<sup>q</sup></I>)

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Polynomial codes

Cited by 104 publications

References 10 publications

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Low-density parity-check codes based on finite geometries: a rediscovery and new results

Asymmetric quantum codes: constructions, bounds and performance

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field &lt;I&gt;GF&lt;/I&gt;(&lt;I&gt;p&lt;sup&gt;q&lt;/sup&gt;&lt;/I&gt;)

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Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field <I>GF</I>(<I>p<sup>q</sup></I>)