CASCUDO, Ignacio; CRAMER, Ronald; MIRANDOLA, Diego; ZÉMOR, Gilles

doi:10.1109/TIT.2015.2393251

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CASCUDO, Ignacio
Aarhus University [Aarhus]

CRAMER, Ronald
Universiteit Leiden = Leiden University
Centrum voor Wiskunde en Informatica [CWI]

MIRANDOLA, Diego
Institut de Mathématiques de Bordeaux [IMB]
Centrum voor Wiskunde en Informatica [CWI]
Universiteit Leiden = Leiden University

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IEEE Transactions on Information Theory. 2015, vol. 61, n° 3, p. 1159-1173

Institute of Electrical and Electronics Engineers

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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.< Leer menos

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Squares of Random Linear Codes

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