In many papers I have been reading, bioengineers are utilizing covering sets from Dan Gordon's La Jolla Covering Repository to design custom Modified Hamming Codes (MHD). These MHDs have custom codeword lengths and desired Hamming weights, that I see some have been using to create codewords outside the (7,4) Hamming code taught in textbooks.
I'm trying to better understand the link between covering designs and linear block codes. Namely, I'm not understanding how (v, k, t) covering design inform the property of a linear block code (ie. what's the Hamming distance? rate? error-detection capacity? error-correcting capacity)? Why can covering designs be used to make a linear block at all?
For clarity, a (v, k, t) covering design is defined below:
A (v,k,t)-covering design is a collection of k-element subsets, called blocks, of {1,2,…,v}, such that any t-element subset is contained in at least one block.