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Spectral distribution of product of pseudorandom matrices formed from binary block codes

Publication ,  Journal Article
Babadi, B; Tarokh, V
Published in: IEEE Transactions on Information Theory
January 24, 2013

Let {\bf A} \in \{-1,1\}^{N-{a} \times n} and {\bf B} \in \{-1,1\}^{N-{b} \times n} be two matrices whose rows are drawn i.i.d. from the codewords of the binary codes {\cal C}a and {\cal C}b of length n and dual distances {d^{\prime}}a and {d^{\prime}}b, respectively, under the mapping 0 \mapsto 1 and 1 \mapsto -1. It is proven that as n \rightarrow \infty with y-{a}:=n/N-{a} \in (0,\infty) and y-{b}:=n/N-{b} \in (0, \infty) fixed, the empirical spectral distribution of the matrix {\bf A} {\bf B}^{\ast }/\sqrt {N-{a} N-{b}} resembles a universal distribution (closely related to the distribution function of the free multiplicative convolution of two members of the Marchenko-Pastur family of densities) in the sense of the Lévy distance, if the asymptotic dual distances of the underlying binary codes are large enough. Moreover, an explicit upper bound on the Lévy distance of the two distributions in terms of ya, yb, {d^{\prime}}a , and {d^{\prime}}b is given. Under mild conditions, the upper bound is strengthened to the Kolmogorov distance of the underlying distributions. Numerical studies on the empirical spectral distribution of the product of random matrices from BCH and Gold codes are provided, which verify the validity of this result. © 1963-2012 IEEE.

Duke Scholars

Published In

IEEE Transactions on Information Theory

DOI

ISSN

0018-9448

Publication Date

January 24, 2013

Volume

59

Issue

2

Start / End Page

970 / 978

Related Subject Headings

  • Networking & Telecommunications
  • 4613 Theory of computation
  • 4006 Communications engineering
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing
 

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Babadi, B., & Tarokh, V. (2013). Spectral distribution of product of pseudorandom matrices formed from binary block codes. IEEE Transactions on Information Theory, 59(2), 970–978. https://doi.org/10.1109/TIT.2012.2223812
Babadi, B., and V. Tarokh. “Spectral distribution of product of pseudorandom matrices formed from binary block codes.” IEEE Transactions on Information Theory 59, no. 2 (January 24, 2013): 970–78. https://doi.org/10.1109/TIT.2012.2223812.
Babadi B, Tarokh V. Spectral distribution of product of pseudorandom matrices formed from binary block codes. IEEE Transactions on Information Theory. 2013 Jan 24;59(2):970–8.
Babadi, B., and V. Tarokh. “Spectral distribution of product of pseudorandom matrices formed from binary block codes.” IEEE Transactions on Information Theory, vol. 59, no. 2, Jan. 2013, pp. 970–78. Scopus, doi:10.1109/TIT.2012.2223812.
Babadi B, Tarokh V. Spectral distribution of product of pseudorandom matrices formed from binary block codes. IEEE Transactions on Information Theory. 2013 Jan 24;59(2):970–978.

Published In

IEEE Transactions on Information Theory

DOI

ISSN

0018-9448

Publication Date

January 24, 2013

Volume

59

Issue

2

Start / End Page

970 / 978

Related Subject Headings

  • Networking & Telecommunications
  • 4613 Theory of computation
  • 4006 Communications engineering
  • 1005 Communications Technologies
  • 0906 Electrical and Electronic Engineering
  • 0801 Artificial Intelligence and Image Processing