On Generalized Minimum Distance Decoding Thresholds for the AWGN Channel

We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney [1], [2] and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo–soft decision decoding can be extended from one threshold T to z > 1 thresholds T1 < · · · < Tz for erasing received symbols with lowest reliability. The resulting technique is widely known as Generalized Minimum Distance decoding. In this paper we provide a means for explicit determination of the optimal threshold locations in terms of minimal decoding error probability. We do this for the one and the general z > 1 thresholds case, starting with a geometric interpretation of the optimal threshold location problem and using an approach from [3].