## 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]**

**[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.]

**[Theorem]**

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)] **

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