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Good codes (those with positive rate $r=k/n$ and positive relative distance $\delta=d/n$) will achieve capacity on $BSC$ (binary symmetric channel) if the codes have lower rates than capacity where positive relative distance is seen. However this requires very long codes to drive the error to reasonably low value.

To achieve an error rate of $e$ if capacity is $C$ then what is the shortest good code that is possible over $BSC$ as a function of $e$? I am just looking for an upper bound and a lower bound.

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  • $\begingroup$ Sorry for my ignorance, but exactly what is the capacity of a code, and what do you mean by "where positive relative distance is seen"? $\endgroup$
    – W-t-P
    Jun 4, 2019 at 15:13
  • $\begingroup$ Capacity is maximum rate supported for negligible error communication over a channel. $\endgroup$
    – Turbo
    Jun 4, 2019 at 20:39
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    $\begingroup$ capacity is a property of the channel not the code $\endgroup$
    – kodlu
    Jun 4, 2019 at 22:00
  • $\begingroup$ Please tell where the discrepancy is. $\endgroup$
    – Turbo
    Jun 4, 2019 at 22:01
  • $\begingroup$ my comment was to clarify capacity in response to @W-t-P $\endgroup$
    – kodlu
    Jun 4, 2019 at 22:03

1 Answer 1

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Upper and lower bounds as well as approximations are given in "Channel Coding Rate in the Finite Blocklength Regime" by Yury Polyanskiy, H. Vincent Poor, and Sergio Verdu.

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