Coding Theory and Design Theory pp 16-34 | Cite as

# The Differential Encoding of Coset Codes by Algebraic Methods

## Abstract

A trellis code is a method of encoding a binary data stream as a sequence of real vectors that are transmitted over a noisy channel. Trellis codes are used in modems designed to achieve data rates of up to 19.2 kb/s on dial-up voice telephone lines. Coset codes are trellis codes based on lattices and cosets. The signal constellation is finite, and signal points are taken from 2*N*-dimensional lattice *L*, with an equal number of points taken from each coset of a sublattice *M*. One part of the input data stream selects cosets of *M* in *L* and the other part selects points from those cosets. An important practical problem is that of channel phase shifts which cause a rotation of every 2-dimensional constituent of a 2*N*-dimensional signal through the same multiple of 90°. We describe the structure of coset codes and an algebraic method of resolving this phase ambiguity.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A.R. Calderbank and J.E. Mazo,
*A new description of trellis codes*, IEEE Trans. Inform. Theory, IT-30 (1984) pp. 784–791.MathSciNetCrossRefGoogle Scholar - [2]A.R. Calderbank, J.E. Mazo, and H.M. Shapiro,
*Upper bounds on the minimum distance of trellis codes*, Bell Syst. Tech. J. 62 (1983) pp. 2617–2646.MathSciNetGoogle Scholar - [3]A.R. Calderbank, J. E. Mazo, and V.K. Wei,
*Asymptotic upper bounds on the minimum distance of trellis codes*, IEEE Trans. Commun., COM-33 No. 4 (1985), pp. 305–309.MathSciNetCrossRefGoogle Scholar - [4]A.R. Calderbank and N.J.A. Sloane,
*Four-dimensional modulation with an eight-state trellis code*, Bell Syst. Tech. J. 64 (1985), pp. 1005–1018.MathSciNetzbMATHGoogle Scholar - [5]A.R. Calderbank and N.J.A. Sloane,
*New trellis codes based on lattices and cosets*, IEEE Trans. Inform. Theory, vol. IT-33 (1987), 177–195.MathSciNetCrossRefGoogle Scholar - [6]G.D. Forney J.,
*Coset codes I: geometrical classification*, to appear in IEEE Trans. Inform. Theory.Google Scholar - [7]G.D. Forney Jr.,
*Coset codes II: Binary lattices and related codes*, submitted to IEEE Trans. Inform. Theory.Google Scholar - [8]G.D. Forney Jr.,
*Coset codes III: Ternary codes, lattices and trellis codes*, submitted to IEEE Trans. Inform. Theory.Google Scholar - [9]G.D. Forney Jr., R.G. Gallager, G.R. Lang, F.M. Longstaff and S.U. Quereschi, Efficient
*modulation for band-limited channels*, IEEE J. Select. Areas Commun., SAC-2 (1984), 632–647.CrossRefGoogle Scholar - [10]
- [11]G.A. Kabatiansky and V.I. Levenshtein,
*Bounds for packing on a sphere and in space*(in Russian), Probl. Peredachi Inform., 14, no. 1 (1978), pp. 3–25; transl. in Probl. Inform. Transmiss., 14, no. 1 (1978), pp. 1–17.Google Scholar - [12]G. Ungerboeck,
*Channel coding with multilevel/phase signals*, IEEE Trans. Inform. Theory, IT-28 (1982), pp. 55–67.CrossRefGoogle Scholar - [13]A. J. Viberbi and J.K. Omura,
*Principles of digital communication and coding*, New York: McGraw-Hill (1979).Google Scholar - [14]L.F. Wei,
*Rotationally invariant convolutional channel coding with expanded signal space - II: Nonlinear codes*, IEEE J. Select. Areas Commun., SAC-2 (1984) pp. 672–686.Google Scholar - [15]L.F. Wei,
*Trellis coded modulation with multidimensional constellations*, to appear in IEEE Trans. Inform. Theory.Google Scholar - [16]E. Zehavi and J.K. Wolf,
*On the performance evaluation of trellis codes*, IEEE Trans. Inform. Theory, IT-33 (1987), pp. 196–202.MathSciNetCrossRefGoogle Scholar