**On powers of 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. 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

**Contact author: **diego at cwi nl

**Available format(s): **PDF | BibTeX Citation

**Version: **20140703:180838 (All versions of this report)

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