# Squares of Random Linear Codes

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.
Domain :

https://hal.archives-ouvertes.fr/hal-01261390
Contributor : Gilles Zémor <>
Submitted on : Monday, January 25, 2016 - 11:37:10 AM
Last modification on : Thursday, January 11, 2018 - 6:21:22 AM

### Identifiers

• HAL Id : hal-01261390, version 1
• ARXIV : 1407.0848

### Citation

Ignacio Cascudo, Ronald Cramer, Diego Mirandola, Gilles Zémor. Squares of Random Linear Codes. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2015, 61 (3), pp.1159--1173. ⟨hal-01261390⟩

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