A new framework for building and decoding group codes

This paper investigates the construction and the decoding of a remarkable set of lattices and codes viewed as group codes: We treat in a unified framework the Leech lattice and the Golay code in dimension 24, the Nebe lattice in dimension 72, the Barnes-Wall lattices, and the Reed-Muller codes. We also present a new family of lattices called the parity lattices. The common aspect of these group codes is that they can be obtained as single parity-check $k$-groups or via the $k$-ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.

[1]  Henry Cohn,et al.  The sphere packing problem in dimension 8The sphere packing problem in dimension 8 , 2016, 1603.04246.

[2]  Yair Be'ery,et al.  Fast decoding of the Leech lattice , 1989, IEEE J. Sel. Areas Commun..

[3]  Patrick Solé,et al.  On the algebraic structure of quasi-cyclic codes I: Finite fields , 2001, IEEE Trans. Inf. Theory.

[4]  Anne Desideri Bracco,et al.  Treillis de codes quasi-cycliques , 2004, Eur. J. Comb..

[5]  Morris Plotkin,et al.  Binary codes with specified minimum distance , 1960, IRE Trans. Inf. Theory.

[6]  Robert L. Griess,et al.  Rank 72 high minimum norm lattices , 2009, 0910.2055.

[7]  N. J. A. Sloane,et al.  New binary codes , 1972, IEEE Trans. Inf. Theory.

[8]  M. Collins,et al.  Four presentatioons of Leech's lattice , 1980 .

[9]  N. J. A. Sloane,et al.  Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice , 1986, IEEE Trans. Inf. Theory.

[10]  Alexander Vardy,et al.  List decoding of polar codes , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[11]  Gabriele Nebe,et al.  Dense lattices as Hermitian tensor products , 2013 .

[12]  J. Lepowsky,et al.  An E8-approach to the Leech lattice and the Conway group☆ , 1982 .

[13]  Thomas M. Thompson.  From error-correcting codes through sphere packings to simple groups , 1983 .

[14]  G. David Forney,et al.  Coset codes-II: Binary lattices and related codes , 1988, IEEE Trans. Inf. Theory.

[15]  Alexander Vardy,et al.  Universal Bound on the Performance of Lattice Codes , 1999, IEEE Trans. Inf. Theory.

[16]  Meir Feder,et al.  Finite-Dimensional Infinite Constellations , 2011, IEEE Transactions on Information Theory.

[17]  Emanuele Viterbo,et al.  Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices , 2012, IEEE Transactions on Communications.

[18]  E. S. Barnes,et al.  Some extreme forms defined in terms of Abelian groups , 1959, Journal of the Australian Mathematical Society.

[19]  Daniele Micciancio,et al.  Efficient bounded distance decoders for Barnes-Wall lattices , 2008, 2008 IEEE International Symposium on Information Theory.

[20]  Léo Ducas,et al.  Polynomial time bounded distance decoding near Minkowski’s bound in discrete logarithm lattices , 2018, IACR Cryptol. ePrint Arch..

[21]  Yanfei Yan,et al.  Polar lattices: Where Arıkan meets Forney , 2013, 2013 IEEE International Symposium on Information Theory.

[22]  Daniel Panario,et al.  Turbo Lattices: Construction and Performance Analysis , 2011, ArXiv.

[23]  Emanuele Viterbo,et al.  A universal lattice code decoder for fading channels , 1999, IEEE Trans. Inf. Theory.

[24]  Gregory Poltyrev,et al.  On coding without restrictions for the AWGN channel , 1993, IEEE Trans. Inf. Theory.

[25]  Martin Bossert,et al.  Soft-decision decoding of Reed-Muller codes as generalized multiple concatenated codes , 1995, IEEE Trans. Inf. Theory.

[26]  Sae-Young Chung,et al.  Sphere-bound-achieving coset codes and multilevel coset codes , 2000, IEEE Trans. Inf. Theory.

[27]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[28]  Gordon R. Lang,et al.  A Leech lattice modem , 1989, IEEE J. Sel. Areas Commun..

[29]  Annika Meyer.  On the number of lattice points in a small sphere and a recursive lattice decoding algorithm , 2013, Des. Codes Cryptogr..

[30]  Shu Lin,et al.  Soft-decision decoding of linear block codes based on ordered statistics , 1994, IEEE Trans. Inf. Theory.

[31]  Richard Parker,et al.  On extremal even unimodular 72-dimensional lattices , 2013, Math. Comput..

[32]  H.-G. Quebbemann.  A construction of integral lattices , 1984 .

[33]  G. Nebe.  A generalisation of Turyn's construction of self-dual codes (Research into Vertex Operator Algebras, Finite Groups and Combinatorics) , 2011 .

[34]  G. David Forney,et al.  Modulation and Coding for Linear Gaussian Channels , 1998, IEEE Trans. Inf. Theory.

[35]  Reginaldo Palazzo Júnior,et al.  A bounded-distance decoding algorithm for lattices obtained from a generalized code formula , 1994, IEEE Trans. Inf. Theory.

[36]  G. David Forney,et al.  Coset codes-I: Introduction and geometrical classification , 1988, IEEE Trans. Inf. Theory.

[37]  Norman P. Secord,et al.  Demodulation of a Gosset lattice code having a spectral null at DC , 1989, IEEE Trans. Inf. Theory.

[38]  G. David Forney,et al.  A bounded-distance decoding algorithm for the Leech lattice, with generalizations , 1989, IEEE Trans. Inf. Theory.

[39]  Alexander Vardy,et al.  Closest point search in lattices , 2002, IEEE Trans. Inf. Theory.

[40]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[41]  Alexander Vardy,et al.  Maximum likelihood decoding of the Leech lattice , 1993, IEEE Trans. Inf. Theory.

[42]  A. Krieg,et al.  On Hermitian theta series and modular forms , 2009 .

[43]  Patrick Solé,et al.  On quintic quasi-cyclic codes , 2008, Discret. Appl. Math..

[44]  G. Nebe,et al.  The unreasonable effectiveness of the tensor product , 2012, 1201.1832.

[45]  N. Sloane,et al.  On the Voronoi Regions of Certain Lattices , 1984 .

[46]  Alexander Vardy.  Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice , 1995, IEEE Trans. Inf. Theory.

[47]  Alexander Vardy,et al.  Generalized minimum distance decoding in Euclidean space: Performance analysis , 1997, IEEE Trans. Inf. Theory.

[48]  Ofer Amrani,et al.  Encoding and Decoding Binary Product Lattices , 2006, IEEE Transactions on Information Theory.

[49]  Ofer Amrani,et al.  The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions , 1994, IEEE Trans. Inf. Theory.

[50]  N. J. A. Sloane,et al.  A fast encoding method for lattice codes and quantizers , 1983, IEEE Trans. Inf. Theory.

[51]  Meir Feder,et al.  Low-Density Lattice Codes , 2007, IEEE Transactions on Information Theory.

[52]  Chris Peikert,et al.  List-Decoding Barnes–Wall Lattices , 2016, computational complexity.

[53]  Ilya Dumer,et al.  Soft-decision decoding of Reed-Muller codes: recursive lists , 2006, IEEE Transactions on Information Theory.

[54]  A. Robert Calderbank,et al.  Quaternary quadratic residue codes and unimodular lattices , 1995, IEEE Trans. Inf. Theory.

[55]  J. Leech.  Notes on Sphere Packings , 1967, Canadian Journal of Mathematics.

[56]  G. David Forney,et al.  Multidimensional constellations. II. Voronoi constellations , 1989, IEEE J. Sel. Areas Commun..

[57]  Shafi Goldwasser,et al.  Complexity of lattice problems - a cryptographic perspective , 2002, The Kluwer international series in engineering and computer science.

[58]  H. F. Mattson,et al.  RESEARCH TO DEVELOP THE ALGEBRAIC THEORY OF CODES. , 1968 .

[59]  Gabriele Nebe,et al.  An even unimodular 72-dimensional lattice of minimum 8 , 2010, 1008.2862.

[60]  Alexander Vardy,et al.  Generalized minimum-distance decoding of Euclidean-space codes and lattices , 1996, IEEE Trans. Inf. Theory.

[61]  Toshiki Matsumine,et al.  Construction D Lattice Decoding and Its Application to BCH Code Lattices , 2018, 2018 IEEE Global Communications Conference (GLOBECOM).

[62]  Loïc Brunel,et al.  Integer low-density lattices based on construction A , 2012, 2012 IEEE Information Theory Workshop.