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enumerative coding for Grassmannian Space

IEEE TRANSACTIONS ON INFORMATION THEORY 2011年第1期57卷 365-374页

作者： Silberstein, Natalia Etzion, Tuvi Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

The Grassmannian space G(q) (n, k) is the set of all k-dimensional subspaces of the vector space F-q(n). Recently, codes in the Grassmannian have found an application in network coding. The main goal of this paper is to present efficient enumerative encoding and decoding techniques for the Grassmannian. These coding techniques are based on two different orders for the Grassmannian induced by different representations of k-dimensional subspaces of F-q(n). One enumerative coding method is based on a Ferrers diagram representation and on an order for G(q) (n, k) based on this representation. The complexity of this enumerative coding is O(k(5/2)(n - k)(5/2) digit operations. Another order of the Grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces. The complexity of the enumerative coding, based on this order, is O(nk(n - k) log n log log n) digit operations. A combination of the two methods reduces the complexity on average by a constant factor.

关键词： enumerative coding Ferrers diagram Grassmannian identifying vector partitions reduced row echelon form

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enumerative coding for line polar Grassmannians with applications to codes

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FINITE FIELDS AND THEIR APPLICATIONS 2017年第Jul.期46卷 107-138页

作者： Cardinali, Ilaria Giuzzi, Luca Univ Siena Dept Informat Engn & Math Via Roma 56 I-53100 Siena Italy Univ Brescia DICATAM Sect Math Via Branze 53 I-25123 Brescia Italy

A k-polar Grassmannian is a geometry having as pointset the set of all k-dimensional subspaces of a vector space V which are totally isotropic for a given non-degenerate bilinear form mu defined on V. Hence it can be regarded as a subgeometry of the ordinary k-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume k = 2 and kt to be a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nice applications;among them, we shall discuss an efficient encoding/decoding/error correction strategy for line polar Grassmann codes of either type. (C) 2017 Elsevier Inc. All rights reserved.

关键词： enumerative coding Polar Grassmannian Linear code

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Universal enumerative coding for Tree Models

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第3期60卷 1387-1411页

作者： Martin, Alvaro Seroussi, Gadiel Weinberger, Marcelo J. Univ Republica Montevideo 11300 Uruguay Hewlett Packard Labs Palo Alto CA 94304 USA

Efficient enumerative coding for tree sources is, in general, surprisingly intricate-a simple uniform encoding of type classes, which is asymptotically optimal in expectation for many classical models, such as FSMs, turns out not to be so in this case. We describe an efficiently computable enumerative code that is universal in the family of tree models in the sense that, for a string emitted by an unknown source whose model is supported on a known tree, the expected normalized code length of the encoding approaches the entropy rate of the source with a convergence rate (K/2)(log n)/n, where K is the number of free parameters of the model family. Based on recent results characterizing type classes of context trees, the code consists of the index of the sequence in the tree type class, and an efficient description of the class itself using a nonuniform encoding of selected string counts. The results are extended to a twice-universal setting, where the tree underlying the source model is unknown.

关键词： Universal coding enumerative coding tree models method of types type classes

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Quasi-enumerative coding of Balanced Run-Length Limited Codes

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IEEE ACCESS 2024年 12卷 39375-39389页

作者： Paluncic, Filip Maharaj, B. T. Univ Pretoria Dept Elect Elect & Comp Engn ZA-0002 Pretoria South Africa

Various methods have been proposed for the construction of balanced run-length limited codes. Amongst these methods is the enumerative coding approach by Kurmaev. The advantage of this approach is that the code has maximum cardinality and thus approaches capacity with increasing codeword length. However, enumerative coding has the disadvantage of becoming prohibitively complex for large codeword lengths. We propose an alternative enumerative coding method that reduces the encoding and decoding complexity. We call it quasi-enumerative coding as it does not follow a strict lexicographic order, but retains a one-to-one mapping between rank and the corresponding balanced run-length limited codeword.

关键词： Codes Encoding Random variables Decoding Symbols Object recognition Media enumerative coding balanced codes run-length limited codes

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Gray Codes and enumerative coding for Vector Spaces

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第1期60卷 271-281页

作者： Schwartz, Moshe Ben Gurion Univ Negev Dept Elect & Comp Engn IL-8410501 Beer Sheva Israel

Gray codes for vector spaces are considered in two graphs: the Grassmann graph, and the projective-space graph, both of which have recently found applications in network coding. For the Grassmann graph, constructions of cyclic optimal codes are given for all parameters. As for the projective-space graph, two constructions for specific parameters are provided, as well some nonexistence results. Furthermore, encoding and decoding algorithms are given for the Grassmannian Gray code, which induce an enumerative-coding scheme. The computational complexity of the algorithms is at least as low as known schemes, and for certain parameter ranges, the new scheme outperforms previously known ones.

关键词： enumerative coding Grassmannian Gray codes projective-space graph

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True-Damage-Aware enumerative coding for Improving NAND Flash Memory Endurance

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IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 2015年第6期23卷 1165-1169页

作者： Li, Jiangpeng Zhao, Kai Ma, Jun Zhang, Tong Shanghai Jiao Tong Univ Shanghai 200240 Peoples R China Rensselaer Polytech Inst Troy NY 12180 USA

This brief presents a technique that can fully exploit the data dependency of flash memory cell damage to improve the program/erase (P/E) cycling endurance of NAND flash memory. The key is to opportunistically leverage data lossless compressibility and utilize the compression gain to realize memory-damage-aware data manipulation to reduce the cycling-induced physical damage. Based upon experiments using commercial sub-22-nm MLC NAND flash memory chips, we show that the proposed design technique can improve the P/E cycling endurance by 50%. We further carried out application-specific integrated circuit design to demonstrate the practical feasibility for implementing the proposed design technique.

关键词： Endurance enumerative coding NAND flash memory

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Permutation (d, k) codes:: Efficient enumerative coding and phrase length distribution shaping

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第7期46卷 2671-2675页

作者： Milenkovic, O Vasic, B Univ Michigan Ann Arbor MI 48109 USA Bell Labs Allentown PA 18103 USA

In this correspondence, we introduce a new enumerative encoding method for (d, k) codes. The encoding algorithm, which is based on enumeration of multiset permutations, is conceptually simpler and computationally less expensive than other algorithms proposed thus far. We also describe a new application of enumerative encoding methods for phrase length distribution shaping of run-length-limited (RLL) sequences. We demonstrate that by reducing the probability of occurrence of long phrases in maxentropic RLL sequences, the frequency of patterns that account for most of the errors in magnetic recording systems can be decreased.

关键词： enumerative coding magnetic recording channels runlength-limited (RLL) codes

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Error propagation assessment of enumerative coding schemes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1999年第7期45卷 2591-2594页

作者： Immink, KAS Janssen, AJEM Inst Expt Math D-45326 Essen Germany Philips Res Labs NL-5656 AA Eindhoven Netherlands

enumerative coding is an attractive algorithmic procedure for translating long source words into codewords and vice versa. The usage of long codewords makes it possible to approach a code rate which is as close as desired to Shannon's noiseless capacity of the constrained channel. enumerative encoding is prone to massive error propagation as a single hit error could ruin entire decoded words. This contribution will evaluate the effects of error propagation of the enumerative coding of runlength-limited sequences.

关键词： enumerative coding error propagation RLL

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Comparison and Optimization of enumerative coding Techniques for Amplitude Shaping

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IEEE COMMUNICATIONS LETTERS 2021年第4期25卷 1231-1235页

作者： Gultekin, Yunus Can van Houtum, Wim J. Koppelaar, Arie G. C. Willems, Frans M. J. Eindhoven Univ Technol TU e Dept Elect Engn EE Informat & Commun Theory Lab ICTLab NL-5612 Eindhoven Netherlands NXP Semicond NV NL-5656 Eindhoven Netherlands

The energy efficiency of enumerative coding algorithms is investigated for amplitude shaping in the context of binary transmission. First, a simple method for the calculation of the amplitude distribution is derived for enumerative sphere shaping (ESS). For ultra-short blocklengths, ESS-which orders sequences lexicographically-is shown to be less energy-efficient than algorithms that use energy-based ordering such as shell mapping. Second, ESS is optimized heuristically such that its energy efficiency is improved. Simulations show that optimized ESS achieves the same error probabilities as energy-based ordering methods for the additive white Gaussian noise (AWGN) channel, even at ultra-short blocklengths.

关键词： Encoding Indexes Signal to noise ratio Optical transmitters OFDM Forward error correction Adaptive optics enumerative coding shell mapping shaping

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Low-Complexity enumerative coding Techniques With Applications to Amplitude Shaping

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IEEE COMMUNICATIONS LETTERS 2021年第1期25卷 33-37页

作者： Gultekin, Yunus Can van Houtum, Wim J. Koppelaar, Arie G. C. Willems, Frans M. J. TUe Dept Elect Engn Signal Proc Syst SPS Grp Informat & Commun Theory ICT Lab NL-5600 Eindhoven Netherlands NXP Semicond NL-5692 Eindhoven Netherlands TUe Dept Elect Engn SPS Grp ICT Lab NL-5600 Eindhoven Netherlands

Practical implementation of enumerative sphere shaping (ESS) is considered. First, an on-the-fly computation method is proposed such that the required storage is significantly decreased, e.g., by a factor of 7 for 8-ary amplitude-shift keying (ASK) at blocklength N = 64 and shaping rate of 1.75 bit/amplitude. Then a sliding window shaping (SWS) architecture is introduced to eliminate the necessity to realize high precision arithmetic operations, and to decrease the computational complexity of ESS. The SWS procedure only requires three 16-bit arithmetic operations per output symbol for 8-ASK.

关键词： Sphere shaping probabilistic shaping enumerative coding

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