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## Averaging bounds for lattices and linear codes (1997)

Venue: | IEEE Trans. Information Theory |

Citations: | 93 - 1 self |

### Citations

12166 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...e of a lattice with fundamental volume . Note that, if is chosen as a set of typical noise vectors (where, this time, “typical” is meant in the formal sense of information theory, e.g., as defined in =-=[20]-=-) then will, for ,converge to the (information-theoretic) differential entropy rate of the noise. (The density of the noise must be sufficiently “nice” so that is Jordan measurable; if is not bounded,... |

557 |
Sphere packings, lattices and groups
- Conway, Sloane
- 1988
(Show Context)
Citation Context ...dered in Section II. In this line of development, Rush and Sloane [14], [15] observed that applying a version of the Varshamov–Gilbert bound to certain linear codes over GF used with “Construction A” =-=[16]-=- proves the existence of lattices with the same asymptotic sphere-packing density as those known earlier from the Minkowski–Hlawka theorem. For large , thisis still the best existence result known for... |

239 |
An introduction to the geometry of numbers
- Cassels
- 1997
(Show Context)
Citation Context ...over a large, usually infinite, class of lattices; in this sense, the Minkowski–Hlawka theorem is random coding and may be regarded as a pre-Shannon result in information theory. The proof by Cassels =-=[10]-=- (as cited in [4]), is actually based on averaging over the set of linear codes over GF , although the connection to coding is not made explicit. From the stated version of the Minkowski–Hlawka theore... |

192 |
Probability of error for optimal codes in a Gaussian channel
- Shannon
- 1959
(Show Context)
Citation Context ...ial error bound. In his second paper [5], de Buda seemed to have proved that lattice codes can even achieve the full channel capacity ,with the same exponential error bounds as Shannon’s random codes =-=[7]-=-. For technical reasons, he considered “thick-shell” shaping rather than spherical shaping; moreover, he assumed a nearest-codeword decoder rather than alattice decoder. However, an error in [5] was r... |

175 | Geometry of Numbers - Gruber, Lekkerkerker - 1987 |

152 |
Packing and Covering
- Rogers
- 1964
(Show Context)
Citation Context ... to a subtle problem that will be discussed at the beginning of Section IV.) All these authors based their lattice results on the following version (due to Hlawka) of the Minkowski–Hlawka theorem [9]–=-=[11]-=-, [12, ch. 3, Theorem 1]: for any Riemann integrable function of bounded support and any positive ,there exists a lattice in with fundamental volume such that (A version of this theorem will be proved... |

136 |
Algebraic-Geometric Codes
- Tsfasman, Vlǎdut¸
- 1991
(Show Context)
Citation Context ...Rush and Sloane did not recognize, however, that (as in Cassels’ proof) the Minkowski–Hlawka theorem itself can be obtained from averaging over Construction A lattices. This approach is summarized in =-=[17]-=-, which emphasizes the role of (versions of) the Minkowski–Hlawka theorem as the lattice analog to the Varshamov–Gilbert bound. The present paper aims at making explicit the mentioned connections betw... |

68 |
On coding without restrictions for the AWGN channel
- Poltyrev
- 1994
(Show Context)
Citation Context ... lattice decoder and not to a nearest-codeword decoder. The main prior work on the information-theoretic limits of lattice codes are two papers by de Buda [4], [5] and a more recent paper by Poltyrev =-=[6]-=-. In his first paper [4], de Buda considered spherically shaped lattice codes with lattice decoding and showed that arbitrarily small error probability Manuscript received July 22, 1994; revised April... |

56 |
Multidimensional constellationsPart II: Voronoi constellations
- Wei
- 1989
(Show Context)
Citation Context ...centered at the origin. For the analysis of such codes, it is common to separate the “coding gain,” which is provided by the lattice, from the “shaping gain,” which stems from the shaping region [2], =-=[3]-=-. In particular, it is usually assumed that the decoder is unaware of the shaping, i.e., it always decodes to the nearest lattice point, whether or not this point lies in .Suchadecoder will be called ... |

47 | New trellis codes based on lattices and cosets
- Calderbank, Sloane
- 1987
(Show Context)
Citation Context ...lattices have become a standard tool for the construction of both block codes and (convolutional-type) trellis codes for the additive white Gaussian noise (AWGN) channel at high signal-to-noise ratio =-=[1]-=-, [2]. Only block codes will be considered in this paper. Such lattice codes consist of the intersection of a lattice (or a translate of a lattice) with a bounded shaping region ,which is typically a ... |

44 | Coset codes-Part I: Introduction and geometrical classification
- Forney
- 1988
(Show Context)
Citation Context ...ces have become a standard tool for the construction of both block codes and (convolutional-type) trellis codes for the additive white Gaussian noise (AWGN) channel at high signal-to-noise ratio [1], =-=[2]-=-. Only block codes will be considered in this paper. Such lattice codes consist of the intersection of a lattice (or a translate of a lattice) with a bounded shaping region ,which is typically a ball ... |

21 |
Corrected proof of de Budas theorem
- Linder, Schlegel, et al.
- 1993
(Show Context)
Citation Context ...he considered “thick-shell” shaping rather than spherical shaping; moreover, he assumed a nearest-codeword decoder rather than alattice decoder. However, an error in [5] was reported by Linder et al. =-=[8]-=-. They were able to fix the problem, at the price of replacing de Buda’s “thick” shells with “thin” shells. Consequently, the corresponding codes lose most of their lattice structure and rather resemb... |

19 |
The upper error bound of a new near-optimal code
- Buda
- 1975
(Show Context)
Citation Context ...ndependent error probability, apply only to a lattice decoder and not to a nearest-codeword decoder. The main prior work on the information-theoretic limits of lattice codes are two papers by de Buda =-=[4]-=-, [5] and a more recent paper by Poltyrev [6]. In his first paper [4], de Buda considered spherically shaped lattice codes with lattice decoding and showed that arbitrarily small error probability Man... |

19 |
Algebraic construction of shannon codes for regular channels
- Delsarte, Piret
- 1982
(Show Context)
Citation Context ...y nonzero element of is contained in the same number, denoted by ,ofcodes from .Itis easily seen that, for fixed and the set of all linear codes over is balanced. The term “balanced” is borrowed from =-=[18]-=- where, however, it was used as a property of a set of affine encoders rather than of linear codes. Being balanced is the key property for most averaging arguments for linear codes such as Varshamov–G... |

14 |
Zur Geometrie der Zahlen
- Hlawka
- 1944
(Show Context)
Citation Context ...tion to a subtle problem that will be discussed at the beginning of Section IV.) All these authors based their lattice results on the following version (due to Hlawka) of the Minkowski–Hlawka theorem =-=[9]-=-–[11], [12, ch. 3, Theorem 1]: for any Riemann integrable function of bounded support and any positive ,there exists a lattice in with fundamental volume such that (A version of this theorem will be p... |

11 |
An improvement to the MinkowskiHlawka bound for packing superballs
- Rush, Sloane
- 1987
(Show Context)
Citation Context ...nkowski–Hlawka theorem” was originally (and sometimes still is) used only for one such result. Results of this type will also be considered in Section II. In this line of development, Rush and Sloane =-=[14]-=-, [15] observed that applying a version of the Varshamov–Gilbert bound to certain linear codes over GF used with “Construction A” [16] proves the existence of lattices with the same asymptotic sphere-... |

7 |
A lower bound on packing density
- Rush
- 1989
(Show Context)
Citation Context ...i–Hlawka theorem” was originally (and sometimes still is) used only for one such result. Results of this type will also be considered in Section II. In this line of development, Rush and Sloane [14], =-=[15]-=- observed that applying a version of the Varshamov–Gilbert bound to certain linear codes over GF used with “Construction A” [16] proves the existence of lattices with the same asymptotic sphere-packin... |

3 |
A Capacity Theorem for Lattice Codes on Gaussian Channels
- Oliveira, B
- 1990
(Show Context)
Citation Context ...e is no longer the intersection of (a translate of) a lattice with a spherical shell. In any case, de Buda’s second paper says nothing about lattice codes with lattice decoding. Magalhäes and Battail =-=[21]-=- also considered lattice codes and derived error exponents. In fact, they seemed to have proved that, even with lattice decoding, the full capacity is achievable. However, a mistake in the proof (the ... |

1 |
Mathematische Probleme,” Arch
- Hilbert
- 1901
(Show Context)
Citation Context ...ersion of the Minkowski–Hlawka theorem, one can derive various existence results for “packing lattices.” In fact, it was the problem of packing -dimensional spheres and other bodies (posed by Hilbert =-=[13]-=-) that motivated Minkowski and Hlawka, and the name “Minkowski–Hlawka theorem” was originally (and sometimes still is) used only for one such result. Results of this type will also be considered in Se... |

1 |
On the basic averaging arguments for linear codes,” in Communications and Cryptography: Two Sides of One Tapestry, (festschrift in honor of James L. Massey on the occasion of his 60th birthday
- Loeliger
- 1994
(Show Context)
Citation Context ...perty for most averaging arguments for linear codes such as Varshamov–Gilbert-type bounds or random coding bounds; in fact, these bounds can be derived from the following lemma (cf., the Appendix and =-=[19]-=-), versions of which are routinely (and usually implicitly) used in coding texts. Lemma 1 (Basic Averaging Lemma): Let be an arbitrary mapping ;let be a balanced set of linear codes over .Thenthe aver... |