On the distribution of rank of a random matrix over a finite field

Cooper, Colin

doi:10.1002/1098-2418(200010/12)17:3/4<197::aid-rsa2>3.0.co;2-k

Cited by 69 publications

(51 citation statements)

References 5 publications

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“…Many of the important results are in the Russian literature and a few of these results are cited here through the English translations, where available. Of the more recent results of which the authors are aware, the papers of Cooper [4,5] and the book [8] are the most relevant to this work. A few of these results are quoted below.…”

Section: Rank Properties Of Random Matricessupporting

confidence: 87%

“…The indistinguishability from the purely random case requires the higher value noted. The conjecture is closely related to previous results of [4,5,8] although the perspective here seems sufficiently distinct to report on it separately.…”

Section: Theorem 32 [5 Theorem 2]mentioning

confidence: 90%

“…The paper of Cooper [5] considers the case of rectangular matrices over an arbitrary finite field F q . The following two theorems, where the notation is adapted to the above, contain perhaps the strongest results of which the authors are aware:…”

Section: Rank Properties Of Random Matricesmentioning

confidence: 99%

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Random Matrices and Codes for the Erasure Channel

Studholme

2008

Algorithmica

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The design of erasure correcting codes and their decoding algorithms is now at the point where capacity achieving codes are available with decoding algorithms that have complexity that is linear in the number of information symbols. One aspect of these codes is that the overhead (number of coded symbols beyond the number of information symbols required to achieve decoding completion with high probability) is linear in k. This work considers a new class of random codes which have the following advantages: (i) the overhead is constant (in the range of 5 to 10), independent of the number of data symbols being encoded (ii) the probability of completing decoding for such an overhead is essentially one (iii) the codes are effective for a number of information symbols as low as a few tens (iv) the only probability distribution required is the uniform distribution. The price for these properties is that the decoding complexity is greater, on the order of k 3/2 . However, for the lower values of k where these codes are of particular interest, this increase in complexity might be outweighed by their advantages. The parity check matrices of these codes are chosen at random as windowed matrices, i.e. for each column an initial starting position of a window of length w is chosen and the succeeding w positions are chosen at random as zero or one. It can be shown that it is necessary that w = O(k 1/2 ) for the probabilistic matrix rank properties to behave as a non-windowed random matrix. The sufficiency of the condition has so far been established by extensive simulation, although other arguments strongly support this conclusion. The properties of the codes described 606 Algorithmica (2010) 56: 605-620 depend heavily on the rank properties of random matrices over finite fields. Known results on such matrices are briefly reviewed and several conjectures needed in the discussion of the code properties, are stated. The likelihood of the validity of the conjectures is supported through extensive experimentation. Mathematical proof of the conjectures would be of great value for their own interest as well of the particular coding application described here.

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Section: Rank Properties Of Random Matricessupporting

confidence: 87%

Section: Theorem 32 [5 Theorem 2]mentioning

confidence: 90%

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Random Matrices and Codes for the Erasure Channel

Studholme

2008

Algorithmica

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“…But this result can only be applied to matrices of infinite size over GF (2). When m = n, the probability that H is nonsingular is found to be quite robust in [14]. Specifically, conditioned on H having no all-zero rows or columns, the probability…”

Section: Introductionmentioning

confidence: 99%

Bounds on the expected rank of sparse linear operator channel matrices

Liu

Mow

2013

2013 19th Asia-Pacific Conference on Communications (APCC)

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The random linear network coding system can be modelled as a linear operator channel (LOC). Among all the LOC models, one of the most interesting models is the sparse LOC model, where the entries of the channel matrix take zero with high probability and any of the nonzero elements in the finite field with equal probabilities. It is well-known that the normalized capacity of LOC is characterized by the rank distribution of the channel matrix. Furthermore, Yang et al. derived upper and lower capacity bounds for LOC in terms of the expected rank of the channel matrix, and the two bounds coincide with the expected rank of the channel matrix as the packet size increases. Therefore, in this paper, we investigate the expected rank of the sparse LOC matrix. By calculating the number of vectors in the null space of the sparse LOC matrix, a lower bound on the expected rank of the sparse matrix is obtained. The linear rank distribution of the sparse LOC matrix is also calculated, which is then used to derive an upper bound on the expected rank of such matrices. Besides, the upper bound can be shown to be exact as the matrix size tends to infinity. Numerical results show that the bounds are rather tight for various matrix sizes and sparsity.

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“…The question of distribution of the number of solutions of a random linear system of equations over the field GF ( ) 3 is considered in [2][3][4][5]. THEOREM.…”

mentioning

confidence: 99%