Circular Trellis Coded Modulation (CTCM) defines a family of (block) trellis codes which use a unique algebraic constraint, imposed on the start state, to produce a strong tail-biting property without the inefficiency of driving the encoder state to zero by using a sequence of input zeroes. From the beginning of CTCM, elements of the Galois field, GF(pm), have served dual roles, labeling both s ystem trellis nodes and valid input symbols. This dual use of field elements facilitates exploitation of the algebraic structure of GF(pm). The system trellis always take a particularly simple and advantageous form (called the pn-fly form) whenever the alphabet of valid input symbols is chosen to be (a coset of) any additive subgroup of the additive group structure of GF(pm). This paperproposes a family of signal mappings that complete the definition of the CTCM system by providing structurally consistent output labels for the trellis edges. The completion of a structural definition greatly facilitates system analysis, especially the (future) geometrically precise construction of a related signal constellation. At the same time, it preserves the possibility of an advantegeous receiver structure.
Circular Trellis Coded Modulation (CTCM) defines a family of (block) trellis codes which use a unique algebraic constraint, imposed on the start state, to produce a strong tail-biting property without the inefficiency of driving the encoder state to zero, using a sequence of input zeroes. Previous papers have investigated specific practical results in the case of binary and 4-ary signaling, using the elements of the Galois field GF(2m) to label the trellis. The `4-fly' has proven to be the central feature of the trellis structure, allowing exceptional performance. The present paper generalizes the 4-fly structure, naturally related to GF(2m), to the case of pj-flies, which support efficient pj-ary signaling using trellises labeled by the field GF(pm), valid both for p equals 2 and for odd characteristic. This opens numerous practical system design options, allows greater flexibility for the transmitter design, and lays the mathematical groundwork required to support a much more systematic and general analysis of the CTCM trellis structure. This paper gives the general definitions and records the first elements of the structural analysis. Detailed knowledge of the trellis structure is key to the minimization of decoder complexity.