The average weight enumerator and the maximum likelihood performance of product codes

El-Khamy, Mostafa

doi:10.1109/wirles.2005.1549650

Cited by 11 publications

(16 citation statements)

References 26 publications

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“…Given a basis {β i } m i=1 in GF(q m ), denote by µ i (γ) ∈ GF(q) the function of γ that represents its expansion associated with basis β i , i.e., the value µ i in (9). For brevity, the function µ i (·) is also applied to a vector y = [y 1 , y 2 , .…”

Section: Characterization Of Expanded Cyclic Codesmentioning

confidence: 99%

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On Expanded Cyclic and Reed–Solomon Codes

Wu¹

2011

IEEE Trans. Inform. Theory

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The paper has a threefold purpose. The first purpose is to present an explicit description of expanded cyclic codes defined in GF(q m ). The proposed explicit construction of expanded generator matrix and expanded parity check matrix maintains the symbol-wise algebraic structure and thus keeps many important original characteristics. The second purpose of this paper is to identify a class of constant-weight cyclic codes. Specifically, we show that a well-known class of q-ary BCH codes excluding the all-zero codeword are constant-weight cyclic codes. Moreover, we show this class of codes achieve the Plotkin bound. The last purpose of the paper is to characterize expanded cyclic codes utilizing the proposed expanded generator matrix and parity check matrix. We characterize the properties of component codewords of a codeword and particularly identify the precise conditions under which a codeword can be represented by a subbasis. Our developments reveal an alternative while more general view on the subspace subcodes of Reed-Solomon codes. With the new insights, we present an improved lower bound on the minimum distance of an expanded cyclic code by exploiting the generalized concatenated structure. We also show that the fixed-rate binary expanded Reed-Solomon codes are asymptotically "bad", in the sense that the ratio of minimum distance over code length diminishes with code length going to infinity. It overturns the prevalent conjecture that they are "good" codes and deviates from the ensemble of generalized Reed-Solomon codes which asymptotically achieves the Gilbert-Varshamov bound.

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Section: Characterization Of Expanded Cyclic Codesmentioning

confidence: 99%

“…It overturns the well-known conjecture that they are "good" codes (cf. [9]) and deviates from the ensemble of generalized Reed-Solomon codes which asymptotically achieves the Gilbert-Varshamov bound [15].…”

Section: Introductionmentioning

confidence: 97%

On Expanded Cyclic and Reed–Solomon Codes

Wu¹

2011

IEEE Trans. Inform. Theory

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“…Product codes were introduced by Elias in 1954 (see references in [1]). They represent the first practical method capable of achieving error-free coding with a nonzero code rate [2].…”

Section: Introductionmentioning

confidence: 99%

“…The erasure recovery performance of product codes based on their minimum distance is unreliable. The weight enumerating function of the product code which fully specifies the performance of the code cannot be determined from the constituent codes without considering the specific interleaver or arrangement [1], [5]. Some researchers assume a random interleaver [8].…”

Section: Introductionmentioning

confidence: 99%

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Enhanced Upper Bound for Erasure Recovery in SPC Product Codes

Muqaibel¹

2009

ETRI J

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Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a closed-form tight upper bound for the postdecoding erasure rate. Closed-form exact expressions are derived for up to seven erasures. Previously published closed-form bounds assumed that all unrecoverable patterns should contain four erasures in a square. Additional non-square patterns are accounted for in the proposed expressions. The derived expressions are verified using exhaustive search. Eight or more erasures are accounted for by using a bound. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or search algorithms, whether exhaustive or novel.

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Progressive algebraic Chase decoding algorithms for Reed–Solomon codes

Zhao

Chen

et al. 2016

IET Communications

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This study proposes a progressive algebraic Chase decoding (PACD) algorithm for Reed-Solomon (RS) codes. On the basis of the received information, 2 η (η > 0) interpolation test-vectors are constructed for the interpolation-based algebraic Chase decoding. A test-vector reliability function is defined to assess their potential for yielding the intended message. The algebraic Chase decoding will then be performed progressively granting priority to decode the testvectors that are more likely to yield the message, and is then terminated once it is found. Consequently, the decoding complexity can be adapted to the quality of the received information. An enhanced-PACD (E-PACD) algorithm is further proposed by coupling the PACD algorithm with the adaptive belief propagation (ABP) decoding. The ABP decoding generates new test-vectors for the PACD algorithm by enhancing the received information. It improves the Chase decoding performance without increasing the decoding complexity exponentially. It is shown that the E-PACD algorithm's complexity can be significantly reduced by utilising the existing interpolation information of the previous Chase decodings'. Our performance evaluations show that the two proposed decoders outperform a number of existing algebraic decoding approaches. Complexity and memory analyses of the PACD algorithm are also presented, demonstrating that this is an efficient RS decoding strategy.

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The average weight enumerator and the maximum likelihood performance of product codes

Cited by 11 publications

References 26 publications

On Expanded Cyclic and Reed–Solomon Codes

On Expanded Cyclic and Reed–Solomon Codes

Enhanced Upper Bound for Erasure Recovery in SPC Product Codes

Progressive algebraic Chase decoding algorithms for Reed–Solomon codes

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