On the weight distribution of random binary linear codes

Linial, Nathan; Mosheiff, Jonathan

doi:10.1002/rsa.20879

Cited by 14 publications

(33 citation statements)

References 12 publications

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“…This paper follows up on the recent result [3] by Linial and Mosheiff. In [3] the authors study the distribution of the number of codewords of a given weight in a random linear code of a given rate, providing tight estimates on the suitably normalized moments of this distribution, up to moments of order o n log n . In this paper we extend the estimates in [3] up to moments of linear order.…”

Section: Introductionmentioning

confidence: 85%

Weight distribution of random linear codes and Krawchouk polynomials

Samorodnitsky¹

2022

Preprint

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For 0 < λ < 1 and n → ∞ pick uniformly at random λn vectors in {0, 1} n and let C be the orthogonal complement of their span. Given 0 < γ < 1 2 with 0 < λ < h(γ), let X be the random variable that counts the number of words in C of Hamming weight i = γn (where i is assumed to be an even integer). Linial and Mosheiff [3] determined the asymptotics of the moments of X of all orders o n log n . In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized k th moment of X is essentially determined by the k th norm of the Krawchouk polynomial K i .

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Section: Introductionmentioning

confidence: 85%

Weight distribution of random linear codes and Krawchouk polynomials

Samorodnitsky¹

2022

Preprint

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“…nN qˇˇk ı for any y P R n and for k as large as possible. For random binary linear code over F n 2 of rate 1´Hppq`δ, 8 Linial and Mosheiff [43] recently managed to characterize the first Opn{ log nq moments of the number of codewords in a Hamming sphere of radius np. It turns out that up to a threshold k 0 , the normalized centered moment…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

List Decoding Random Euclidean Codes and Infinite Constellations

Zhang

Vatedka

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints P and N respectively, the list decoding capacity is equal to 1 2 log P N . Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity.

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“…Sidel'nikov [4] posed a question about the difference between the weight distribution spectrum of the whole vector space and the weight distribution spectrum of a random linear code. Mathematical expectation of the difference between the spectrum of random code and the spectrum of vector space (more precisely, the spectrum that is proportional to the spectrum of vector space) is estimated in [3] and [9] for binary vector spaces. As expected, a sequence of random linear codes has a uniform weight spectrum.…”

Section: Introductionmentioning

confidence: 99%

On weight spectrum of linear codes

N.¹

2021

Preprint

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We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors from codes with weight 1 grows to infinity. We find an example of a sequence of linear codes such that the dimension of the code is the half of the codelength but it has not a uniform weight spectrum. This example generates eigenfunctions of the Fourier transform with minimal support and partial covering sets. Moreover, we generalize some MacWilliams-type identity.

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On the weight distribution of random binary linear codes

Cited by 14 publications

References 12 publications

Weight distribution of random linear codes and Krawchouk polynomials

Weight distribution of random linear codes and Krawchouk polynomials

List Decoding Random Euclidean Codes and Infinite Constellations

On weight spectrum of linear codes

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