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

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.

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## 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

### Citation

*IEEE Transactions on Information Theory*,

*59*(2), 970–978. https://doi.org/10.1109/TIT.2012.2223812

*IEEE Transactions on Information Theory*59, no. 2 (January 24, 2013): 970–78. https://doi.org/10.1109/TIT.2012.2223812.

*IEEE Transactions on Information Theory*, vol. 59, no. 2, Jan. 2013, pp. 970–78.

*Scopus*, doi:10.1109/TIT.2012.2223812.

## Published In

## DOI

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## 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