Varshamov-Gilbert lower bound

For block codes it is possible to get the Varshamov-Gilbert lower bound which defines the possibility of codes existence with both parameters R _code and d _min. The asymptotic form (for long codes) of this boundlooks like:

R _code ³ 1 – H (d _min /n),(6.2)

where H (x) – binary entropy.

dependence of Varshamov-Gilbert lower bound for binary codes is shown on figure6.1 (curve «Varshamov-Gilbert lower bound»). The bound guarantees existence of the codes which performances correspond to points arranged at least on a curve (or above it). Search of the codes ensuring the given minimum distance d _min and high enough rate R _code at n→∞, ensuring at the same time a possibility of algorithms decoding realization with low complexity is one of important problem of the theory of coding.