Paper 2014/520
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
Metadata
- Available format(s)
-
PDF
- 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
- 2014-07-03: received
- See all versions
- Short URL
- https://ia.cr/2014/520
- License
-
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}
}
