Ellipsoidal lists and maximum-likelihood decoding

We study an interrelation between the coverings generated by linear ( )–codes and complexity of their maximum-likelihood (ML) decoding. First, discrete ellipsoids in the Hamming spaces are introduced. These ellipsoids represent the sets of most probable error patterns that need to be tested in soft-decision ML decoding. We show that long linear ( )-codes surrounded by ellipsoids of exponential size 2 can cover the whole space . Then it is proven that ML decoding of most long ( )-codes needs only about 2 most probable error patterns to be tested on any quantized memoryless channel. Finally, ML decoding complexity is bounded from above by2 . This substantially reduces general trellis complexity 2 .