Computing the Stopping Distance of a Tanner Graph Is NP-Hard

Krishnan, K. Murali; Shankar, Priti

doi:10.1109/tit.2007.896864

Cited by 51 publications

(29 citation statements)

References 14 publications

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“…Much like rank-deficient sets, it is very challenging to give a full characterization of dead-end sets of an arbitrary group generation matrix A [23]. Therefore, it is also challenging to determine the decoding guarantee of extended grouping based on A.…”

Section: Remarkmentioning

confidence: 99%

“…Because, to the best of our knowledge, there is no known efficient algorithm for finding rank-deficient or dead-end sets of any size [23], we have difficulty providing theoretical results of error rates for any given parameters. Therefore, we present theoretical error rates only for the small target collection size n = 25 and mainly use simulation to evaluate different schemes.…”

Section: A Study Setupmentioning

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: Remarkmentioning

confidence: 99%

Section: A Study Setupmentioning

confidence: 99%

Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Tonguz

2013

IEEE Trans. Commun.

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“…; T g, relative to the super-channel determined by . By (11), for rate R = S s=1 C s 04, the error probability averaged over the ensemble of codes PrfŴ() 6 = W g is not larger than 2 for each decoder , for all large enough block lengths T . Here > 0 is the parameter that specifies the typical set A T .…”

Section: S S=1mentioning

confidence: 99%

Signaling Over Arbitrarily Permuted Parallel Channels

Willems

Gorokhov

2008

IEEE Trans. Inform. Theory

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“…Algorithms to continue the decoding procedure in case a nonempty stopping set is reached are presented in [17], [1]. It has been shown in [11], [14] that determining the sizes of stopping sets is NP-hard.…”

Section: Introductionmentioning

confidence: 99%

“…where the last property follows from the observation that any set S of size jSj > n 0 k contains the support of a nonzero codeword as the Let s denote the smallest size of a nonempty stopping set (and thus the smallest size of a dead-end set), i.e., s = minfi 1 : S i > 0g = minfi 0 : D i > 0g: (11) The number s is called the stopping distance for the parity-check matrix H H H in [19]. For any parity-check matrix H H H of a binary linear [n; k; d] block code C, it holds that the stopping set enumerator satisfies…”

Section: Introductionmentioning

confidence: 99%

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

Weber

Abdel-Ghaffar

2008

IEEE Trans. Inform. Theory

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Abstract-The performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying Tanner graph, or, equivalently, the parity-check matrix representing the code. In this correspondence, we introduce the notion of dead-end sets to explicitly demonstrate this dependency. The choice of the parity-check matrix entails a tradeoff between performance and complexity. We give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity-check matrices. Further, we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator.

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Computing the Stopping Distance of a Tanner Graph Is NP-Hard

Abstract: Two decision problems related to the computation of stopping sets in Tanner graphs are shown to be NP-complete. It follows as a consequence that there exists no polynomial time algorithm for computing the stopping distance of a Tanner graph unless P=NP.

Cited by 51 publications

References 14 publications

Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Using the Chinese Remainder Theorem for the Grouping of RFID Tags

Signaling Over Arbitrarily Permuted Parallel Channels

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

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