Boundaries of block codes parameters

The problem of the coding theory is the search of codes which at given block length n and rate R _code provides a maximum of code distance d _min. Limits of these parameters are defined by the code boundaries.

6.1 Hamming upper bound

The conclusion of the upper bound is based on reasons of spherical packing (bound of spherical packing). At given minimum distance between the allowed code words d _min. The greatest rate can be reached, if the spheres surrounding each word will be most densely packed. volume of each sphere is equal and the number of spheres (number of code words) is equal 2 ^k. For best codethe total quantity of spheres and number of all possible words 2 ⁿ should coincide. Equality is reached for densely packed (perfect) codes. area of each code word represents sphere with radius (d _min – 1)/2, and these areas of such codes not being crossed densely fill with themselves all n -dimensional space of code words. The inequality from here follows:

.

After simple transformations it is possible to receive obvious expression for rate of the perfect code:

. (6.1)

The dependence of Hamming upper bound is shown on figure 6.1 (curve «Hamming upper bound»). Hamming bound is fair both for linear and nonlinear codes.