Block Codes February 21, 2006

A code with fixed code length, n, is called a block code. For such a block code, there are bounds on the number of codes M, given a code length n and minimum distance d (and hence a bound on the error-detecting and correcting capabilities).

[Hamming Bound]

Hamming bound

[Gilbert-Varshamov Bound]

Gilbert-Varshamov bound

[At some point in the past, I was looking at the possibility of changing the shape of a density function by re-shuffling its support. At that point, it looked like the following theorem [see Baylis] would be useful. Turned out not to be what I needed once I understood the problem better.]

Performing a positional permutation on the words of a code does not change its minimum distance.

[Plotkin Bound]
The Hamming and Singleton (not shown above) bounds give bounds on M for some arbitrary n and d. If willing to constrain d, the Plotkin gives a tighter (can we really say that ?) bound on M for the case when d > n/2, which given the dependence on number of correctable errors on d, will be useful for when we need to correct a largen number of errors.

[Theorem (Plotkin bound)]

Plotkin bound

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