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Some Mysterious Sequences Associated with LDPC Codes

McEliece, Robert J. and Sweatlock, Sarah L. (2007) Some Mysterious Sequences Associated with LDPC Codes. In: Sequences, Subsequences, and Consequences. Lecture Notes in Computer Science. No.4893. Springer , Berlin, pp. 153-161. ISBN 978-3-540-77403-7.

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One of the most important research areas in coding theory is weight enumeration. This is a large subject, but the basic problem is easily stated: determine or estimate the weight enumerator (B_0, ..., B_n) for an (n,k) binary linear code, specified by a (n − k)×n parity-check matrix H with entries from GF(2). Here B_i = #{c∈GF(2)^n:Hc^T = 0, wt(c = i)}, where wt(c) is the weight of the vector c. If the number of codewords is large, the logarithmic weight enumerator, i.e., (1/n logB_0,…,1/n log B_n) is more convenient. If a code belongs to a family of codes which share similar properties, the log-weight enumerator may approach a limiting function called the spectral shape: 1/n log (B_(⌊θn⌋)) → E(θ), 0 < θ < 1. In modern coding theory, H is usually very large and very sparse, e.g., the row and column sums are bounded as n → ∞. The corresponding codes are called low density parity-check codes. Often we are faced with large collections, or ensembles, of long LDPC codes, which share similar properties, in which case it may be difficult to find the spectral shape of an individual member of the ensemble, but relatively easy to calculate the ensemble average.

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Additional Information:© 2007 Springer-Verlag Berlin Heidelberg. This research was supported by the Lee Center for Advanced Networking and the Sony Corporation.
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Caltech Lee Center for Advanced NetworkingUNSPECIFIED
Sony CorporationUNSPECIFIED
Series Name:Lecture Notes in Computer Science
Issue or Number:4893
Record Number:CaltechAUTHORS:20200204-105926568
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101113
Deposited By: Tony Diaz
Deposited On:04 Feb 2020 19:19
Last Modified:04 Feb 2020 19:19

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