Statistical characteristics of weight spectra of random linear codes over $\mathrm{GF}(p)$

For a random uniform subcode of fixed linear code over the finite field $\mathbf F_p$ its weight spectrum is considered. Formulas for the first two moments of the weight spectrum elements and estimates for the minimal nonzero weight distribution of subcode elements are derived in terms of weight spectrum of the code. Formulas for the first two moments and the weight distribution of sum of two independent random vectors having fixed weights are given also.