Edge local complementation and equivalence of binary linear codes

Abstract. We describe an operation to dynamically adapt the structure of the Tanner graph used during iterative decoding. Codes on graphsmost importantly, low-density parity-check (LDPC) codes-exploit randomness in the structure of the code. Our approach is to introduce a similar degree of controlled randomness into the operation of the messagepassing decoder, to improve the performance of iterative decoding of classical structured (i.e., non-random) codes for which strong code properties are known. We use ideas similar to Halford and Chugg (IEEE Trans. on Commun., April 2008), where permutations on the columns of the parity-check matrix are drawn from the automorphism group of the code, Aut(C). The main contributions of our work are: 1) We maintain a graph-local perspective, which not only gives a low-complexity, distributed implementation, but also suggests novel applications of our work, and 2) we present an operation to draw from Aut(C) such that graph isomorphism is preserved, which maintains desirable properties while the graph is being updated. We present simulation results for the additive white Gaussian noise (AWGN) channel, which show an improvement over standard sum-product algorithm (SPA) decoding.

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“…The operation of edge local complementation (ELC) [10][11][12], also known as Pivot, is a local operation on a simple graph. Fig.…”

Section: Edge Local Complementationmentioning

confidence: 99%

“…The matrices H i in this orbit are the set of structurally different parity-check matrices for the same code, as discussed in the Introduction. We briefly mention that all information sets of C may be enumerated by traversing this orbit of G [11].…”

Section: Edge Local Complementationmentioning

confidence: 99%

Adaptive Soft-Decision Iterative Decoding Using Edge Local Complementation

Knudsen

Riera

Parker

et al.

Coding Theory and Applications

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“…Depending on H, the sequence e may not be unique, so equivalent sequences may be derived from P and P ′ . Proof: ELC generates the entire orbit [15], and in particular all systematic parity-check matrices for the corresponding code, so such a sequence e must exist. Since a systematic basis for a (dual) code is uniquely defined (up to row permutations) by its parity set, the information set (i.e., the P -part of H) is a function of the parity set.…”

Section: A Minimum-length Elc Sequence Between Two Structuresmentioning

confidence: 99%

“…This paper describes the pseudorandom use of a simplegraph operation known as edge-local complementation (ELC) [14,15] to improve the performance of SPA decoding. One advantage of ELC-enhanced SPA decoding is the locality argument; diversity is achieved by local graph action, and so is well-suited to the local actions that constitute the SPA.…”

Section: Introductionmentioning

confidence: 99%

Random Edge-Local Complementation with Applications to Iterative Decoding of High-Density Parity-Check Codes

Knudsen

Riera²,

Danielsen³

et al. 2012

IEEE Trans. Commun.

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Abstract-We describe the application of edge-local complementation (ELC) to a Tanner graph associated with a binary linear code, C. Various properties of ELC are described, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), and the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upperbounded. ELC generates all systematic parity-check matrices (the orbit) of the code, so WB-ELC facilitates a restriction to lowweight matrices of this orbit. We propose using ELC and WB-ELC as a source of diversity, to improve iterative soft-input softoutput decoding of high-density parity-check (HDPC) codes, with the sum-product algorithm (SPA). A motivation of ELC-enhanced SPA decoding is locality; that diversity is achieved by local graph action, and is well suited to the local actions that constitute the SPA and allows for parallel software implementation. Simulation data on the error-rate performance of the proposed SPA-ELC and SPA-WBELC iterative decoding algorithms is shown for several HDPC codes. A gain is reported over SPA decoding, and over a recently proposed algorithm to decode HDPC codes using permutations from Aut(C). ELC-enhanced decoding extends the scope of iterative decoding to codes with trivial Aut(C).

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“…It is possible to both classify and/or enumerate this class given a classification and/or enumeration of all self-dual codes, coupled with a method to classify and/or enumerate all rowcolumn inequivalent systematic parity check matrices for each code. One way of performing this last task is to generate all edge-local complementation (ELC) orbits (Danielsen and Parker, 2008), to within re-labelling of vertices, for the bipartite graph associated with each distinct self-dual code of size n. For each of self-dual and anti-self-dual, enumeration would then be realised by summing the orbit sizes and then multiplying the result by 2 n/2−1 , and classification would be realised by listing each member in the union of orbits. Each member of such a list would then be a RM(2, n) coset leader for a coset of 2 n/2−1 self-dual and 2 n/2−1 anti-self-dual quadratic Boolean functions.…”

Section: Proof: the Dual Of A Maiorana-mcfarland Bent Functionmentioning

confidence: 99%

Self-dual bent functions

Carlet

Danielsen

Parker

et al. 2010

IJICOT

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A bent function is called self-dual if it is equal to its dual. It is called anti-self-dual if it is equal to the complement of its dual. A spectral characterization in terms of the Rayleigh quotient of the Sylvester Hadamard matrix is derived. Bounds on the Rayleigh quotient are given for Boolean functions in an odd number of variables. An efficient search algorithm based on the spectrum of the Sylvester matrix is derived. Primary and secondary constructions are given. All self-dual bent Boolean functions in ≤ 6 variables and all quadratic such functions in 8 variables are given, up to a restricted form of affine equivalence.

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Edge local complementation and equivalence of binary linear codes

Cited by 16 publications

References 17 publications

Adaptive Soft-Decision Iterative Decoding Using Edge Local Complementation

Adaptive Soft-Decision Iterative Decoding Using Edge Local Complementation

Random Edge-Local Complementation with Applications to Iterative Decoding of High-Density Parity-Check Codes

Self-dual bent functions

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