# Error-correcting rate is misleading

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 $1-\epsilon$. Is this because we artificially restrict the channel to change only "symbols" rather than bits, or is it because the code is actually better?

• Why would you restrict yourself to binary codes if your transmission medium/technology can handle many more? – Raphael Mar 24 '12 at 12:19
• @Raphael It would help if you could justify your point with a few practical examples of real-life technologies handling non-binary symbols and post that as an answer. – Mohammad Alaggan Mar 24 '12 at 16:23
• @M.Alaggan: I'm no expert on this; I figure if you can encode 0/1 on a wave carrier, you can encode many more symbols, too, transmitting more information by time interval. It would surprise me if modern technology would not do this (think code-multiplexing) but I can not name a concrete example. – Raphael Mar 24 '12 at 16:37
• @Raphael I think you are right, current digital-communication channels DO work with larger symbols, but not more than, say, 256-bit per symbol (which is quite rare for wireless, but may be common for cables). But the symbol-size is limited to very small sizes, and cannot (practically) grow at will. – Ran G. Mar 24 '12 at 18:12