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As the title states, why are long block lengths commonly assumed or used in channel coding proofs?

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    $\begingroup$ What reading have you done? Which proof or proofs are you thinking of? What have you tried -- have you tried cranking through the proof without that assumption to see what went wrong? Without more details/specifics, it is hard to answer your question. We generally prefer detailed, well-researched questions; a one-sentence question is often not a good fit for this site. $\endgroup$ – D.W. Dec 14 '14 at 0:21
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One reason is that this allows a small fixed overhead per block. With a sufficiently large block, the effect of this becomes minimal (it's amortized out over the length of the block).

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Shannon's source coding theorem shows that you can encode at the rate of entropy. Shannon's channel theorem shows that you can transmit data at the rate of channel capacity. In both theorems, both quantities – source entropy and channel capacity – are only approached in the limit of a large block size. If you limit the size of your block, the theorems are not true anymore. The only way to get a clean theory is by allowing the block size to grow unboundedly.

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