In coding theory, \'how good a code is\' means how many channel errors can be co

Question

In coding theory, 'how good a code is' means how many channel errors can be corrected, or better put, the maximal noise level that the code can deal with.

In order to get better codes, the codes are designed using a large alphabet (rather than binary one). And then, the code is good if it can deal with a large rate of erroneous "symbols".

Why isn't this consider cheating? I mean, shouldn't we only care about what happens when we "translate" each symbol into a binary string? The "rate of bit error" is different than the rate of "symbol error". For instance, the rate of bit-error cannot go above 1/2 while (if I understand this correctly), with large enough alphabet, the symbol-error can go up to 1??. Is this because we artificially restrict the channel to change only "symbols" rather than bits, or is it because the code is actually better?

Explanation / Answer

Many widely used codes for binary data are concatenated codes, which are composed by using two error-correcting codes. The inner code is over a binary alphabet, and the outer code is over an alphabet whose symbols correspond to the codewords of the inner code. This allows you to use the superior power of larger alphabet sizes to encode binary messages without "cheating".

The standard definition of minimum distance is a natural one to use when considering concatenated codes, as well as in the theory of codes over large alphabet sizes. It would only be "cheating" if you used these numbers to compare a binary code with a large-alphabet code that encodes binary input without using an inner code as well; coding theorists are clever enough not to do this (and I believe that since concatenated codes were invented, large-alphabet codes have often been used along with an inner code, but large-alphabet codes are also very good for correcting error in bursty channels such as CDs, since a large number of consecutive bit errors will only affect a few "symbols").

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