Ellipsoidal lists and maximum-likelihood decoding

We study an interrelation between the coverings generated by linear (n,k)-codes and complexity of their maximum-likelihood (ML) decoding. First , discrete ellipsoids in the Hamming spaces E₁ⁿ 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 (n,k)-codes surrounded by ellipsoids of exponential size 2^n-k can cover the whole space E₂ⁿ. Then it is proven that ML decoding of most long (n,k)-codes needs only about 2^n-k most probable error patterns to be tested on any quantized memoryless channel. Finally, ML decoding complexity is bounded from above by 2^k(n-k)n/. This substantially reduces the general trellis complexity 2^min{n-k,k}