Squares of Random Linear Codes


Given a linear code $C$ , one will define the $d$ th power of $C$ because the span of all componentwise merchandise of $d$ parts of $C$ . An influence of $C$ might quickly fill the whole space. Our purpose is to answer the following question: does the square of a code usually fill the whole area? We tend to offer a positive answer, for codes of dimension $k$ and length roughly $(1/2)k^2$ or smaller. Moreover, the convergence speed is exponential if the distinction $k(k+1)/2-n$ is at least linear in $k$ . The proof uses random coding and combinatorial arguments, along with algebraic tools involving the precise computation of the quantity of quadratic varieties of a given rank, and the quantity of their zeros.

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