### Squares of Random Linear Codes

Ignacio Cascudo, Ronald Cramer, Diego Mirandola, and Gilles Zémor

##### Abstract

Given a linear code $C$, one can define the $d$-th power of $C$ as the span of all componentwise products of $d$ elements of $C$. A power of $C$ may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code typically'' fill the whole space? We give a positive answer, for codes of dimension $k$ and length roughly $\frac{1}{2}k^2$ or smaller. Moreover, the convergence speed is exponential if the difference $k(k+1)/2-n$ is at least linear in $k$. The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros.

Note: Final version, to appear on IEEE Transactions on Information Theory

Available format(s)
Category
Foundations
Publication info
Preprint. MINOR revision.
Keywords
Error-correcting codes
Contact author(s)
diego @ cwi nl
History
2015-01-14: last of 2 revisions
See all versions
Short URL
https://ia.cr/2014/520

CC BY

BibTeX

@misc{cryptoeprint:2014/520,
author = {Ignacio Cascudo and Ronald Cramer and Diego Mirandola and Gilles Zémor},
title = {Squares of Random Linear Codes},
howpublished = {Cryptology ePrint Archive, Paper 2014/520},
year = {2014},
note = {\url{https://eprint.iacr.org/2014/520}},
url = {https://eprint.iacr.org/2014/520}
}

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