Multiple-phase codes for detection without carrier phase reference

We consider the construction and analysis of linear block codes for M-ary phase-shift keying that can be decoded without carrier phase synchronization. Under these circumstances, the function that has a significant impact on performance is the noncoherent distance, analogous to the Euclidean distance for the coherent case. The major difficulty in constructing and analyzing such codes lies in the fact that the noncoherent distance is not a true metric. For this reason, prior work mainly relies on numerical approaches to search for good codes and to determine the corresponding minimum noncoherent distance. We first present a theorem that links the noncoherent distance with the Euclidean and Lee (1958) distances. This theorem allows us to construct good codes and determine their minimum noncoherent distances analytically. Based on this theorem, we classify the codes whose duals consist of a cyclic group. These codes are of minimum redundancy. We further investigate codes with the flavor of Hamming and shortened Hamming codes. Many of these new codes provide significantly larger coding gains than previously known codes. Linear codes derived from code-division multiple-access (CDMA) sequences are considered as well. These codes in general provide rather large coding gains. Finally, an algorithm is introduced that can be appended to any suboptimal decoding technique to enhance the performance.