Results on Parity-Check Matrices With Optimal Stopping And/Or Dead-End Set Enumerators

Weber, Jos H.; Abdel-Ghaffar, Khaled

doi:10.1109/tit.2007.915923

Cited by 31 publications

(29 citation statements)

References 22 publications

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“…It has been demonstrated in [21] and [22] that decoding failures of iterative message passing decoding over erasure channels are characterized by certain structures, called deadend sets, of a parity-check matrix of a code. However, for the purpose of our exposition, we use a slightly modified version of dead-end sets, which is given below.…”

Section: B Decoding Guarantee and Error Rate Of Extended Groupingmentioning

confidence: 99%

“…The main difference between Definition 5 and that in [21] is the need to investigate the submatrix A em of the group generation matrix A. This is because information of some rows of A may not be available upon decoding.…”

Section: Remarkmentioning

confidence: 99%

“…This is because information of some rows of A may not be available upon decoding. Therefore, similarly to Remark 3, if, in Definition 5, A em is replaced with A, the smallest size of dead-end sets of A is equal to the stopping distance of a linear code with A as its parity-check matrix [21], [22].…”

Section: Remarkmentioning

confidence: 99%

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Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Tonguz

2013

IEEE Trans. Commun.

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Abstract-In this paper, we propose a novel scheme for the design of grouping of radio-frequency identification (RFID) tags, based on the Chinese remainder theorem (CRT). Grouping allows verifying the integrity of a collection of objects without the requirement for accessing external systems, and can be extended to identify missing objects. Motivated by the redundancy property of the Chinese remainder representation, we propose grouping of RFID tags by the CRT. The proposed scheme not only provides designated decoding guarantees, but also offers flexibility in constructing group generation matrices. We also characterize the key objects needed to study decoding guarantees of grouping and their extended counterpart, called rank-deficient and deadend sets, respectively, which enable theoretical analyses of error rates. The two key objects are related to the minimum and stopping distances of a linear code, respectively. As such, the characterization offers direct connection with coding theory that helps with the understanding of the verification/identification problems being studied. Theoretical and simulation results are presented, demonstrating that the proposed scheme is an efficient approach to the design of grouping of RFID tags.Index Terms-Grouping of radio-frequency identification (RFID) tags, integrity, missing-object identification, Chinese remainder theorem (CRT).

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Section: B Decoding Guarantee and Error Rate Of Extended Groupingmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Tonguz

2013

IEEE Trans. Commun.

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“…Clearly, the exhaustive decoder is an optimal decoder with highest decoding complexity. Let Furthermore, Weber and Abdel-Ghaffar [1] show that…”

Section: Introductionmentioning

confidence: 98%

“…The performance of optimal decoding for binary linear codes over a binary erasure channel (BEC) depends on the incorrigible sets of this code [1]. Alike the weight distribution of a linear code for error detection over a binary symmetric channel, the so-called incorrigible set distribution characterizes the probability of unsuccessful decoding of linear codes under optimal decoding over the BEC.…”

Section: Introductionmentioning

confidence: 99%

Incorrigible set distributions and unsuccessful decoding probability of linear codes

Jiang

Xia

Liu

et al. 2013

2013 IEEE International Symposium on Information Theory

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On a binary erasure channel (BEC) with erasing probability , the performance of a binary linear code is determined by the incorrigible sets of the code. The incorrigible set distribution (ISD) {Ii} n i=0 enumerates the number of incorrigible sets with size i of the code. The probability of unsuccessful decoding under optimal decoding for the code could be formulated by the ISD and . In this paper, we determine the ISDs for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some ReedMuller codes or their shortening or puncturing versions. Then, we show that the probability of unsuccessful decoding under optimal decoding for any binary linear code is monotonously non-decreasing on in the interval [0, 1].

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Decoding error probability of random parity-check matrix ensemble over the erasure channel

Chan,

Fu,

Xiong

2024

Des. Codes Cryptogr.

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In this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

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Results on Parity-Check Matrices With Optimal Stopping And/Or Dead-End Set Enumerators

Cited by 31 publications

References 22 publications

Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Incorrigible set distributions and unsuccessful decoding probability of linear codes

Decoding error probability of random parity-check matrix ensemble over the erasure channel

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