Using Pseudocodewords to Transmit Information
The linear programming decoder will occasionally output fractional-valued sequences that do not correspond to binary codewords - such outputs are termed nontrivial pseudocodewords. Feldman et al. have demonstrated that it is precisely the presence of nontrivial pseudocodewords that prevents the linear programming decoder from attaining maximum-likelihood performance. The purpose of this paper is to cast a positive light onto these nontrivial pseudocodewords by turning them into beneficial additions to our codebooks. Specifically, we develop a new modulation scheme, termed insphere modulation, that is capable of reliably transmitting both codewords and pseudocodewords. The resulting non-binary, non-linear codebooks have higher spectral efficiencies than the binary linear codes from which they are derived and in some cases have the same or slightly better block error rates. In deriving our new modulation scheme we present an algorithm capable of computing the insphere of a polyhedral cone - which loosely speaking is the largest sphere contained within the cone. This result may be of independent mathematical interest.