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I think I got what you mean. Please correct me if I misunderstand your question. So, I think "$0 < \vert M_i\vert < m$" in the description of property $\Gamma$ should read "$0 < \vert M_i\vert \leq k …

I don't quite follow the first paragraph of your question. But reading the rest of your post, if you only need a set $S$ of $N$ $m$-dimensional vectors over the finite field of some small order in whi …

If you regard each row of your desired matrix as a sequence of bipolar signals and each column as a time frame, then, with one additional condition that the matrix is circulant, what you're asking bec …

I doubt there is a particular algorithm worth mentioning for avoiding $2 \times 2$ all-one submatrices (or better known as $4$-cycles in the context of LDPC codes) in parity-check matrices which is sp …

For the generalization in the first direction, the $p$-rank of the incidence matrix $N$ of an $S(2,k,v)$ is lower bounded by the dimension of the Steinberg module: $$\operatorname{rank}_2(N)(\operato …

I guess you exclude trivial MDS codes, generalized Reed-Solomon codes, and MDS codes that can be obtained by code extension. If you exclude them all, there are still a bunch of MDS codes. In general, …