5,333 research outputs found

    Multiple Packing: Lower Bounds via Infinite Constellations

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    We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 N>0 and LZ2 L\in\mathbb{Z}_{\ge2} . A multiple packing is a set C\mathcal{C} of points in Rn \mathbb{R}^n such that any point in Rn \mathbb{R}^n lies in the intersection of at most L1 L-1 balls of radius nN \sqrt{nN} around points in C \mathcal{C} . Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive the best known lower bounds on the optimal density of list-decodable infinite constellations for constant LL under a stronger notion called average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04408 and arXiv:2211.0440

    Multiple Packing: Lower and Upper Bounds

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    We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let N>0 N>0 and LZ2 L\in\mathbb{Z}_{\ge2} . A multiple packing is a set C\mathcal{C} of points in Rn \mathbb{R}^n such that any point in Rn \mathbb{R}^n lies in the intersection of at most L1 L-1 balls of radius nN \sqrt{nN} around points in C \mathcal{C} . We study the multiple packing problem for both bounded point sets whose points have norm at most nP\sqrt{nP} for some constant P>0P>0 and unbounded point sets whose points are allowed to be anywhere in Rn \mathbb{R}^n . Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive various bounds on the largest possible density of a multiple packing in both bounded and unbounded settings. A related notion called average-radius multiple packing is also studied. Some of our lower bounds exactly pin down the asymptotics of certain ensembles of average-radius list-decodable codes, e.g., (expurgated) Gaussian codes and (expurgated) spherical codes. In particular, our lower bound obtained from spherical codes is the best known lower bound on the optimal multiple packing density and is the first lower bound that approaches the known large LL limit under the average-radius notion of multiple packing. To derive these results, we apply tools from high-dimensional geometry and large deviation theory.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04408 and arXiv:2211.0440

    Multiple Packing: Lower Bounds via Error Exponents

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    We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. Multiple packing is a natural generalization of the sphere packing problem. For any N>0 N>0 and LZ2 L\in\mathbb{Z}_{\ge2} , a multiple packing is a set C\mathcal{C} of points in Rn \mathbb{R}^n such that any point in Rn \mathbb{R}^n lies in the intersection of at most L1 L-1 balls of radius nN \sqrt{nN} around points in C \mathcal{C} . We study this problem for both bounded point sets whose points have norm at most nP\sqrt{nP} for some constant P>0P>0 and unbounded point sets whose points are allowed to be anywhere in Rn \mathbb{R}^n . Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing a curious inequality which relates the list-decoding error exponent for additive white Gaussian noise channels, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We also derive various bounds on the list-decoding error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.Comment: The paper arXiv:2107.05161 has been split into three parts with new results added and significant revision. This paper is one of the three parts. The other two are arXiv:2211.04407 and arXiv:2211.0440

    Density of Spherically-Embedded Stiefel and Grassmann Codes

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    The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

    Downlink Steered Space-Time Spreading Assisted Generalised Multicarrier DS-CDMA Using Sphere-Packing-Aided Multilevel Coding

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    This paper presents a novel generalised Multi-Carrier Direct Sequence Code Division Multiple Access (MC DS-CDMA) system invoking smart antennas for improving the achievable performance in the downlink, as well as employing multi-dimensional Sphere Packing (SP) modulation for increasing the achievable diversity product. In this contribution, the MC DS-CDMA transmitter considered employs multiple Antenna Arrays (AA) and each of the AAs consists of several antenna elements. Furthermore, the proposed system employs both time- and frequency- (TF) domain spreading for extending the achievable capacity, when combined with a novel user-grouping technique for reducing the effects of Multiuser Interference (MUI). Moreover, in order to further enhance the system’s performance, we invoke a MultiLevel Coding (MLC) scheme, whose component codes are determined using the so-called equivalent capacity based constituent-code rate-calculation procedure invoking a 4-dimensional bit-to-SP-symbol mapping scheme. Our results demonstrate an approximately 3.8 dB Eb/N0 gain over an identical throughput scheme dispensing with SP modulation at a BER of 10?5

    Perfect, strongly eutactic lattices are periodic extreme

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    We introduce a parameter space for periodic point sets, given as unions of mm translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension d8d\leq 8 and d=24d=24.Comment: 20 pages, 1 table; some corrections, incorporated referee suggestion

    Turbo Detection of Symbol-Based Non-Binary LDPC-Coded Space-time Signals using Sphere Packing Modulation

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    A recently proposed space-time signal construction method that combines orthogonal design with sphere packing, referred to here as (STBC-SP), has shown useful performance improvements over Alamouti’s conventional orthogonal design. As a further advance, non-binary LDPC codes have been capable of attaining substantial performance improvements over their binary counterparts. In this paper, we demonstrate that the performance of STBC-SP systems can be further improved by concatenating sphere packing aided modulation with non-binary LDPC codes and performing symbolbased turbo detection. We present simulation results for the proposed scheme communicating over a correlated Rayleigh fading channel. At a BER of 10?6, the proposed symbolbased turbo-detected STBC-SP scheme was capable of achieving a coding gain of approximately 26.6dB over the identical throughput 1 bit/symbol uncoded STBC-SP benchmarker scheme. The proposed scheme also achieved a coding gain of approximately 3dB at a BER of 10?6 over a recently proposed bit-based turbo-detected STBC-SP benchmarker scheme
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