Line codes generated by finite Coxeter groups
Ezio Biglieri Emanuele Viterbo
cs.IT math.IT












 











 
 
















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    Line codes generated by finite Coxeter groups
    
        Ezio Biglieri, Life Fellow, IEEE,
    Emanuele Viterbo, Fellow, IEEE
    The work of E. B. was supported by Project TEC2015-66228-P, while that of
    E. V.  was supported by ARC under Grant Discovery Project No. DP160101077.
    Preliminary versions of this manuscript were presented in <cit.>.Ezio Biglieri is with the Electrical Engineering Department, UCLA, and with Departament TIC, Universitat Pompeu Fabra, Barcelona, Spain (email:  e.biglieri@ieee.org). Emanuele Viterbo is with the Department of Electrical and Computer Systems Engineering, Monash University,
    Melbourne, Australia (email:  emanuele.viterbo@monash.edu).
    
        December 30, 2023
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    Using an algebraic approach based on the theory of Coxeter groups, we design, and describe the performance of, a class of line codes for parallel transmission of b bits over b+1 wires that admit especially simple encoding and decoding algorithms. A number of designs are exhibited, some of them being novel or improving on previously obtained codes.
    
    
    
    Line coding, group codes for the Gaussian channels, permutation modulation, Coxeter groups.
    
    
    
    
    
    § INTRODUCTION AND MOTIVATION OF THE WORK
    
    In this paper we describe the design of vector line codes allowing an especially simple maximum-likelihood (ML) detection procedure. This consists of a linear transformation of the vector received at the output of an additive white Gaussian noise (AWGN) channel, followed by a binary slicer. The design is based on the selection of a subset of a permutation modulation (PM) codebook being the direct product of binary antipodal
    signaling schemes, and hence having a geometrical representation in the form of a multidimensional  orthotope (or hyper-rectangle). The encoder can also be implemented as a linear transformation of the source (binary) vector.
    
    Transmission on parallel wireline links (as those used to interconnect integrated circuits, or a television set to a set-top box) is affected by disturbances placing a number of constraints on the design of the signaling scheme. The key problem here is the design of line codes allowing the transmission of b bits over w ⩾ b wires and using a codebook W subject to some constraints to be detailed later. The general scheme is shown in Fig. <ref>.
    
    h!0.9figure1bis General scheme of vector coding.figure1bis
    Here, b binary information symbols ± 1 are input in parallel to the (w,b) line encoder, which is a one-to-one map from {± 1 }^b to the codebook W ⊆ℝ^w. This encoder outputs
    a code vector w with w real components, which is added to white Gaussian noise to
    obtain vector y. Vector y is processed by a detector whose output is an estimate b of the information vector.
    
    Fig. <ref> illustrates the two basic circuits for wired binary communications:
    (a) unipolar signaling and (b) differential signaling.
