In this paper, parameterized Gallager's first bounding technique (GFBT) is presented by introducing nested Gallager regions, to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The three well-known bounds, namely, the sphere bound (SB) of Herzberg and Poltyrev, the tangential bound (TB) of Berlekamp, and the tangential-sphere bound (TSB) of Poltyrev, are generalized to general codes without the properties of geometrical uniformity and equal energy. When applied to the binary linear codes, the three generalized bounds are reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in the literatures.

arXiv admin note: substantial text overlap with arXiv:1202.0592