Connections between Construction D and related constructions of lattices

Kositwattanarerk, Wittawat; Oggier, Frédérique

doi:10.1007/s10623-014-9939-3

Cited by 31 publications

(45 citation statements)

References 14 publications

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“…Denote by Λ C the smallest lattice that contains Γ C . Kositwattanarerk and Oggier [13] give a condition that if satisfied guarantees that Construction C will provide a lattice which coincides with Construction D. Theorem 1.…”

Section: Definition 1 (Construction A)mentioning

confidence: 99%

“…Remark 3. Note that with the assumption that C = C 1 × C 2 × · · · × C L , i.e., Γ C = Γ C , it follows that S ⊆ C is equivalent to C 1 ⊆ C 2 ⊆ · · · ⊆ C L and the chain is closed under Schur product [13]. Indeed, i) S ⊆ C ⇒ C 1 ⊆ C 2 ⊆ · · · ⊆ C L and the chain is closed under Schur product: we know that S ⊆ C for any pair c,c of codewords, so we take in particularc = c and it follows that…”

Section: Geometric Uniformity and Latticeness Of Construction Cmentioning

confidence: 99%

See 1 more Smart Citation

Multilevel Constructions: Coding, Packing and Geometric Uniformity

Bollauf

Zamir

Costa

2019

IEEE Trans. Inform. Theory

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Lattice and special nonlattice multilevel constellations constructed from binary codes, such as Construction C, have relevant applications in mathematics (sphere packing) and communication problems (multi-stage decoding and efficient vector quantization). In this work, we explore some properties of Construction C, in particular its geometric uniformity. We then propose a new multilevel construction, motivated by bit interleaved coded modulation, that we call Construction C . We explore the geometric uniformity and minimum distance properties of Construction C , and discuss its potential superior packing efficiency with respect to Construction C.

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Section: Definition 1 (Construction A)mentioning

confidence: 99%

Section: Geometric Uniformity and Latticeness Of Construction Cmentioning

confidence: 99%

Multilevel Constructions: Coding, Packing and Geometric Uniformity

Bollauf

Zamir

Costa

2019

IEEE Trans. Inform. Theory

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“…If the family satisfies the Schur product condition, namely, c 1 * c 2 ∈ C i+1 for all codewords c 1 , c 2 ∈ C i , where the '*' operator is the coordinate-wise (Schur) product c 1 * c 2 = (c 1 ) i · (c 2 ) i i∈[n] , then the code-formula forms a lattice (see [21]) and we denote it by L( C i m−1 i=0 ).…”

Section: Upper and Lower Bounds For Testing Specific Lattice Familiesmentioning

confidence: 99%

“…In particular, it is easy to see that code formula lattices of height m ≥ log n in which each of the constituent codes C i has minimum Hamming distance Ω(n) give asymptotically good families of lattices [14,9]. The code formula lattice constructed from a family of codes that satisfies the Schur-product condition is equivalent to the lattice constructed from the same family of codes by Construction D [22,9,21]. Construction-D lattices are primarily used in communication settings, e.g.…”

Section: Upper and Lower Bounds For Testing Specific Lattice Familiesmentioning

confidence: 99%

Local Testing of Lattices

Chandrasekaran¹,

Cheraghchi²,

Gandikota³

et al. 2018

SIAM J. Discrete Math.

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Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography.Apart from establishing the conceptual foundations of lattice testing, our results include the following:1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds.2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21).In this work we initiate the study of local testability for membership in point lattices, a class of infinite algebraic objects that form subgroups of Z n . Lattices are well-studied in mathematics, physics and computer science due to their rich algebraic structure [9]. Algorithms for various lattice problems have directly influenced the ability to solve integer programs [10,23,17]. Recently, lattices have found applications in modern cryptography due to attractive properties that enable efficient computations and security guarantees [28,26,31,32]. Lattices are also used in practical communication settings to encode data in a redundant manner in order to protect it from channel noise during transmission [12].A point lattice L ⊂ R n of rank k and dimension n is specified by a set of linearly independent vectors b 1 , . . . , b k ∈ Z n known as a basis, for some k ≤ n. If k = n the lattice is said to have full rank. The set L is defined to be the set of all vectors in R n that are integer linear combinations of the basis vectors, i.e., L :Lattices are the analogues over Z of linear error-correcting codes over a finite field F, which are generated as F-linear combinations of a linearly independent set of basis vectors b 1 , . . . , b k ∈ F n .Given a basis for a lattice L, we are interested in testing if a given input t ∈ R n belongs to L, or is far from all points in L by querying a small number of coordinates of t. We emphasize that this setting does not limit the computational space or time in pre-processing the lattice as well as the queried coordinates. The main goal is to design a tester that queries only a small number of coordinates of the input. MotivationInteger Programming. Lattices are the fundamental structures underlying integer programming problems. An integer programming problem (IP) is specified by a constraint matrix A ∈ Z n×m , a vector b ∈ R n . The goal is to verify if there exists an integer solution to the system Ax = b, x ≥ 0. Although IP is NP-complete [18], its instances are solv...

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“…A wide range of applicable lattices in digital communications have been treated including the well-known root lattices [ 21 ] , Construction A [22] and Construction Ac [23] . Unfortunately, the presence of random coding with Construction A is a main drawback making such a lattice ensemble too far from any practicality.…”

Section: Constructions Of Lattices and Practical Lattice Codesmentioning

confidence: 99%

Study Status and Prospect of Lattice Codes in Wireless Communication

Duan

Zhao

et al. 2015

2015 Fifth International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC)

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Lattice codes can achieve the capacity of additive white Gaussian noise channel with and without power construction. It is well known that lattice codes can provide a classical information theoretic way to obtain achievable rate and performance gains for point-to-point Gaussian channels. The coding scheme based on lattice codes can be used as building blocks for practical applications, such as point-topoint communication systems, Gaussian networks and MIMO communication systems etc. More importantly, lattice codes can be used to investigate new link adaptive coding and modulation schemes and design sparse code multiple access codebooks. So, lattice codes could be a potential candidate for 5G communication. This paper provides a comprehensive review of the state-of-the-art of lattice codes. It first describes the theory of lattice codes. Then, a comprehensive survey to easily understand channel capacity, error exponents and practical lattice codes is outlined. The application of nested lattice codes is also presented for lossy source coding, dirty paper channel coding and relay communication. Finally, some unresolved issues and ongoing developments related to further improvements of lattice codes are highlighted.

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Connections between Construction D and related constructions of lattices

Cited by 31 publications

References 14 publications

Multilevel Constructions: Coding, Packing and Geometric Uniformity

Multilevel Constructions: Coding, Packing and Geometric Uniformity

Local Testing of Lattices

Study Status and Prospect of Lattice Codes in Wireless Communication

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