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This is an extended comment that may be too long for the comment section. If I understand your question correctly, the linear codes you described are automatically trivial. Define $\pi_{c}$ to be th …

As mentioned in Hao Chen's answer, what you're looking for seems to be a good mixed code. There don't seem to be many papers on this. But apparently the following paper gives the best known general up …

It is called perfect hash families in the design theory and computer science literature. A perfect hash family PHF$(N; k, v, t)$ is an $N \times k$ array on $v$ symbols with $v \geq t$, where for eve …

Here's a long comment that can't be put in the comment section: If you know the location of errors in a binary system, and if errors are just bit flips, you can just flip those erroneous guys again. …

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, …

Your question is very similar to the extended idea of erasure-resilient codes discussed here: Y. M. Chee, C. J. Colbourn, A. C. H. Ling, Asymptotically optimal erasure-resilient codes for large disk …

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 …

Before answering your question, not every Hamming code is equivalent to some cyclic code. For instance, the ternary $[4,2,3]_3$ Hamming code (aka the tetracode) is not equivalent to any cyclic code. …

My knowledge is limited when it comes to general combinatorial packings. But the kind of problem OP described is important in coding theory, where the difference between proving the existence of a goo …

Generally speaking, understanding the decoding error probability of an LDPC code is a very difficult problem. Among major channels that have extensively been studied, as far as I know, binary erasure …

Since you asked a reference, the early access pre-edit version of a paper that addresses exactly this problem you're considering just appeared in the IEEE Transactions on Information Theory: K. A. S. …

So, the supposedly the sharpest one among all known bounds is somehow poorer than the bound you learn in Coding Theory 101 if the alphabet size $q$ approaches infinity. I think the reason you find it …

My previous answer is already too long, and this is too much to include in a comment. But I found a paper that studies the problem you asked here: R. M. Roth, G. Seroussi, Bounds for binary codes wit …

I think you're assuming $x \not= y$ when you say "for any $x, y \in S$." In any case, your question seems like a mix of coding theory and design theory. If you find the case when $a = b = \frac{n}{2} …

Edit: since I only implicitly answered your question in the title (i.e., hot topics in error coding codes related to interesting math), I think one of the current hottest ones is quantum error-correct …