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 $y=p\mathrm{/}n\in (0,1)$ is fixed. We show that as $n\to \mathrm{\infty }$, its empirical spectral distribution converges to the Marchenko-Pastur law at a rate at least of the order $n^{-1\mathrm{/}4}$ 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.