Random Vectors of Bounded Weight and Their Linear Dependencies

Let μ be a probability distribution on a vector space V . When m vectors u1, . . . , um are drawn from μ, how likely are they to be linearly dependent? How is the dimension of their linear span distributed? Such questions have been addressed in a number of papers (e.g. [1],[2],[3],[6],[7]). Our work is motivated by problems in coding theory, and we address these problems in the following context: Here V = F q , the n-dimensional vector space over the field of order q and the distribution μ is uniform over the set of vectors with Hamming weight ≤ w. Let Mm×n be a random matrix whose rows u1, . . . , um are sampled independently from μ. We investigate two associated random variables: (i) The rank of such a random matrix M , (ii) The cardinality of kernel(M). Finally, we consider the distribution of random sums of such randomly chosen vectors u1, . . . , um. Of particular interest to us is to find the least Hamming weight w where the restriction on the vectors’ weights hardly matters. Namely, where the answers become nearly identical with the case w = n, in which vectors are selected uniformly from the entire space. Institute of Computer Science, Hebrew University, Jerusalem 91904, Israel. Work supported in part by grants from the Binational Israel-US Science Foundation and the Israel Academy of Science. Institute of Computer Science, Hebrew University, Jerusalem 91904, Israel.