On the Group Randomness of 0-1 Real Sequences from Binary Linear Codes

In this paper, we study the group randomness of 0-1 real sequences derived from a binary linear code by investigating the spectral behaviour of a suitable normalization of the Gram matrix of a $p \times n$ random matrix whose rows are uniformly drawn from those 0-1 real sequences, where $y=p/n \in (0,1)$ is fixed.

We show that as $n \to \infty$, its empirical spectral distribution converges to the Marchenko-Pastur law at a rate at least of the order $n^{-1/4}$ with high probability, and the fluctuation of its largest eigenvalue is asymptotically Gaussian with mean $p+1+y$ and variance $4y$, provided that the dual distance of the code is at least 5.