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Title:
Squares of Random Linear Codes
Authors: Ignacio Cascudo, Ronald Cramer, Diego Mirandola, 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.
ePrint: https://eprint.iacr.org/2014/520
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