Pseudocyclic maximum-distance-separable codes

Krishna, A.; Sarwate, D.V.

doi:10.1109/18.53751

Cited by 134 publications

(56 citation statements)

References 2 publications

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“…Let C be a λ-constacyclic code with defining set Z = C q+2 ∪C q+5 ∪· · ·∪C q−1+3δ , where 2 ≤ δ ≤ 2(q−2) 3 . Since 1 = ord 3n (q 2 ), from Lemma 4 in [25] (the readers also can see Theorem 4.2 in [36]), the parity check matrix H of C can be denoted as…”

Section: Proposition 4 (Quantum Singleton Bound) (Seementioning

confidence: 99%

“…Now, we can assume that C 1 is a λ-constacyclic code over F q 2 with defining set Z 1 = C q−1+3δ . From Lemma 4 in [25], the parity check matrix H 1 of C 1 can be denoted as H 2,q−1+3δ = 1 η q−1+3δ η 2(q−1+3δ ) · · · η (n−1)(q−1+3δ ) .…”

Section: Proposition 4 (Quantum Singleton Bound) (Seementioning

confidence: 99%

“…Some families of good quantum codes have been constructed in [4,9,24,26,27,40,45]. Nowadays, some scholars studied some families of constacyclic codes in [5][6][7][8]10,20,25,39]. As we know, constacyclic codes contain cyclic codes and negacyclic codes.…”

Section: Introductionmentioning

confidence: 99%

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Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

Chen

Huang

et al. 2015

Linear Algebra and its Applications

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Quantum maximal-distance-separable (MDS) codes are an important class of quantum codes. Recently, many scholars utilize constacyclic codes to construct some quantum MDS codes. In this paper, several new families of optimal asymmetric quantum codes and optimal quantum convolutional codes are constructed by using constacyclic codes. Moreover, these quantum codes constructed in this paper are different from the ones in the literature.

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Section: Proposition 4 (Quantum Singleton Bound) (Seementioning

confidence: 99%

Section: Proposition 4 (Quantum Singleton Bound) (Seementioning

confidence: 99%

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Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

Chen

Huang

et al. 2015

Linear Algebra and its Applications

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“…The BCH bound for Constacyclic codes (see [3,15] for instance) establishes that if C is an α-constacyclic code of length n over F q with generator polynomial g(x) and if g(x) has the elements {β 1+ri |0 ≤ i ≤ d−2} as roots, (β is a primitive rn-th root of unity), then the minimum distance d C of C satisfies d C ≥ d. Recall that given an arbitrary [n, k, d] q linear code C, the Singleton bound asserts that d ≤ n − k + 1. If the parameters of C satisfy d = n − k + 1, the code is called maximum-distance-separable (MDS) or optimal.…”

Section: Constacyclic Codesmentioning

confidence: 99%

On optimal constacyclic codes

Guardia¹

2016

Linear Algebra and its Applications

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In this paper we construct new families of maximum-distance-separable (MDS) convolutional codes derived from classical constacyclic codes. Additionally, we also construct new families of almost MDS block and convolutional codes. Most of the new convolutional codes constructed here are unit-memory and have degree γ = 2. Moreover, based on a different method, we construct families of asymmetric quantum codes known in the literature. All these results are mainly derived from the investigation of suitable properties on cyclotomic cosets of such codes.

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“…We show that the matrix H (q−1,q+1) t consists of all zero and circulant permutation matrices of order q −1. Furthermore, we establish a connection between the code from H (q−1,q+1) t and a constacyclic MDS code (also known as a pseudocyclic MDS code [12]). It is shown that a code from H (q−1,q+1) is identified as a code derived from a (q + 1, 2, q) constacyclic MDS code in a manner similar to the derivation of the RS-based LDPC codes presented by Chen et al [2] and Djurdjevic et al [3].…”

Section: Introductionmentioning

confidence: 98%

High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes

Kamiya

2005

Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.

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This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Here, for codes contained in this class, the exact minimum-distance and a lower bound on the multiplicity of the minimum-weight codewords are presented. It is shown that the lower bound on the multiplicity provides a very accurate indication of the bit error performance at moderate and high signal-to-noise ratios, and thus error-floor behavior can be easily predicted. Also discussed is a class consisting of codes from circulant permutation matrices. An explicit formula for the rank of the parity-check matrix is presented for these codes. Furthermore, it is shown that each of these codes can be identified as a code constructed from a constacyclic maximum distance separable (MDS) code in a similar manner to the RS-based LDPC codes presented by Chen et al. and Djurdjevic et al. Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes.

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Pseudocyclic maximum-distance-separable codes

Cited by 134 publications

References 2 publications

Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

On optimal constacyclic codes

High-rate quasi-cyclic low-density parity-check codes derived from finite affine planes

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