Gallager first bounding technique for the performance evaluation of maximum-likelihood decoded linear binary block codes

Exact bit or frame error rate expressions for most communication systems are either too complex or unlikely to exist in nice closed forms. A good alternative is to bound the performance measure by tight enough lower and upper bounds. Many tight upper bounds on the error probability of binary codes are based on the so-called Gallager´s first bounding technique (GFBT). In this method, Gallager splits the error probability to the joint probability of error and noise residing in a region ℜ (here referred to as the Gallager region) plus joint probability of error and noise residing in the complement of ℜ (also referred to as regions of many and few errors, respectively); where ℜ is a volume around the transmitted codeword. A comprehensive study of a number of upper bounds on the error probability of ML decoding of binary codes based on GFBT is provided. For some bounds, their applicability to other schemes and channels is also pointed out and argued.