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The Fractality of Polar and Reed-Muller Codes.

ABSTRACT: The generator matrices of polar codes and Reed-Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed-Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals.

SUBMITTER: Geiger BC

PROVIDER: S-EPMC7512269 | biostudies-literature | 2018 Jan

REPOSITORIES: biostudies-literature

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The Fractality of Polar and Reed-Muller Codes.

Geiger Bernhard C BC

Entropy (Basel, Switzerland) 20180117 1

The generator matrices of polar codes and Reed-Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed-Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those ro ...[more]

PMID: 33265155

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Project description:Hybridization is observed frequently in birds, but often it is not known whether the hybrids are fertile and if backcrossing occurs. The breeding ranges of the great reed warbler (Acrocephalus arundinaceus) and the clamorous reed warbler (A. stentoreus) overlap in southern Kazakhstan and a previous study has documented hybridization in a sympatric population. In the present study, we first present a large set of novel microsatellite loci isolated and characterised in great reed warblers. Secondly, we evaluate whether hybridization in the sympatric breeding population has been followed by backcrossing and introgression.We isolated 181 unique microsatellite loci in great reed warblers. Of 41 loci evaluated, 40 amplified and 30 were polymorphic. Bayesian clustering analyses based on genotype data from 23 autosomal loci recognised two well-defined genetic clusters corresponding to the two species. Individuals clustered to a very high extent to either of these clusters (admixture proportions ≥ 0.984) with the exception of four previously suggested arundinaceus-stentoreus hybrid birds that showed mixed ancestry (admixture proportions 0.495-0.619). Analyses of simulated hybrids and backcrossed individuals showed that the sampled birds do not correspond to first-fourth-generation backcrosses, and that fifth or higher generation backcrosses to a high extent resemble 'pure' birds at this set of markers.We conclude that these novel microsatellite loci provide a useful molecular resource for Acrocephalus warblers. The time to reach reproductive isolation is believed to be very long in birds, approximately 5 Myrs, and with an estimated divergence time of 2 Myrs between these warblers, some backcrossing and introgression could have been expected. However, there was no evidence for backcrossing and introgression suggesting that hybrids are either infertile or their progeny inviable. Very low levels of introgression cannot be excluded, which still may be an important factor as a source of new genetic variation.

Dataset Information

The Fractality of Polar and Reed-Muller Codes.

Publications

The Fractality of Polar and Reed-Muller Codes.

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