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I've encountered a system where I need to construct a sort of quasi block code:

We want to encode a symbol $s$ from a finite-sized alphabet $\mathcal{S}$ using $N$ segments of information. The $i^{th}$ segment must come from the alphabet $A_i$.

We are tasked with designing a code for this encoder. The code will be a 1-1 function: $$C:\mathcal{S}\rightarrow A_1\times A_2 \times \dots \times A_N$$

What is the code with the highest possible minimum hamming distance between elements in its range?

Note that this is not the same as a "variable-length block code."

Note that if $|S|=2^m,$ and $\underset{i}{\max} |A_i|=2^n$ we could just examine binary (m,n) block codes, and take the best one that varies little enough that it can be re-written as a function onto $A_1 \times \dots \times A_N$.

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    $\begingroup$ This sounds like a MDS code (i.e. one which meets the singleton bound) if I'm reading the question correctly. $\endgroup$ – Batman Dec 15 '14 at 14:40
  • $\begingroup$ It could be considered some form of Distributed Source Coding en.wikipedia.org/wiki/Distributed_source_coding $\endgroup$ – enthdegree Dec 15 '14 at 15:48
  • $\begingroup$ The question is unclear to me. What constraints do you have on $n$ and on $A_i$? Why not just taking any code $F^k \to F^m$ and interpreting each symbol in $F^m$ as $n$ symbols over some alphabets $A_i$? Please try to define the order of quantifiers... $\endgroup$ – Ran G. Jun 14 '15 at 0:38

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