Let \(\alpha \) be a finite field, let \(n\) and \(k \ge 1\) be natural numbers, and fix a nonzero codeword \(x \in \alpha ^{k}\) together with an integer \(d \gt 0\). Regard an \(n \times k\) matrix \(G\) over \(\alpha \) as chosen uniformly at random among all \(\left\lvert \alpha \right\rvert ^{nk}\) such matrices, so that the image \(Gx\) is a codeword of length \(n\). Then the probability that \(Gx\) has Hamming weight strictly less than \(d\) is at most the fraction of all codewords of length \(n\) that lie in the Hamming ball of radius \(d-1\) about the all-zero codeword:

Let \(\alpha \) be a finite nonempty field, let \(n\) and \(k\) be natural numbers with \(k \ge 1\), and let \(d\) be a positive integer. Regard an \(n \times k\) matrix \(G\) over \(\alpha \) as a map sending a message \(x \in \alpha ^{k}\) to the length-\(n\) codeword \(Gx\). Then the number of such matrices \(G\) for which some nonzero message \(x\) has \(\mathrm{wt}(Gx) \lt d\) is at most

where \(\mathrm{wt}\) denotes Hamming weight and \(B_{d-1}(\mathbf0)\) is the Hamming ball of radius \(d-1\) about the all-zero codeword of length \(n\).

Let \(\alpha \) be a finite field with \(q = \left\lvert \alpha \right\rvert \) elements, and let \(n, k, d\) be natural numbers. Write \(B_{d-1}(\mathbf{0})\) for the Hamming ball of radius \(d-1\) about the all-zero codeword in \(\alpha ^n\), and suppose

Then there exists an \(n \times k\) generator matrix \(G\) over \(\alpha \) such that for every nonzero message \(x \in \alpha ^k\), the codeword \(Gx \in \alpha ^n\) has Hamming weight at least \(d\).