## Cryptology ePrint Archive: Report 2014/520

Squares of Random Linear Codes

Ignacio Cascudo and Ronald Cramer and 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.

Category / Keywords: foundations / Error-correcting codes

Date: received 3 Jul 2014, last revised 14 Jan 2015

Contact author: diego at cwi nl

Available format(s): PDF | BibTeX Citation

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

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