arXiv subjects

Hsiao-feng Lu

Publications and source records attributed to Hsiao-feng Lu.

8 recordsLinked to original sources

arXiv2016

On the MacWilliams Identity for Classical and Quantum Convolutional Codes

The weight generating functions associated with convolutional codes (CCs) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing- Luerssen and Schneider in the case of minimal encoders, and generalized by Forney. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity weight adjacency matrices of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum convolutional codes.

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arXiv2015

Analysis and Practice of Uniquely Decodable One-to-One Code

In this paper, we consider the so-called uniquely decodable one-to-one code (UDOOC) that is formed by inserting a "comma" indicator, termed the unique word (UW), between consecutive one-to-one codewords for separation. Along this research direction, we first investigate several general combinatorial properties of UDOOCs, in particular the enumeration of the number of UDOOC codewords for any (finite) codeword length. Based on the obtained formula on the number of length-n codewords for a given UW, the per-letter average codeword length of UDOOC for the optimal compression of a given source statistics can be computed. Several upper bounds on the average codeword length of such UDOOCs are next established. The analysis on the bounds of average codeword length then leads to two asymptotic bounds for sources having infinitely many alphabets, one of which is achievable and hence tight for a certain source statistics and UW, and the other of which proves the achievability of source entropy rate of UDOOCs when both the block size of source letters for UDOOC compression and UW length go to infinity. Efficient encoding and decoding algorithms for UDOOCs are also given in this paper. Numerical results show that the proposed UDOOCs can potentially result in comparable compression rate to the Huffman code under similar decoding complexity and yield a smaller average codeword length than that of the Lempel-Ziv code, thereby confirming the practicability of UDOOCs.

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arXiv2014

Algebraic Codes and a New Physical Layer Transmission Protocol for Wireless Distributed Storage Systems

In a wireless storage system, having to communicate over a fading channel makes repair transmissions prone to physical layer errors. The first approach to combat fading is to utilize the existing optimal space-time codes. However, it was recently pointed out that such codes are in general too complex to decode when the number of helper nodes is bigger than the number of antennas at the newcomer or data collector. In this paper, a novel protocol for wireless storage transmissions based on algebraic space-time codes is presented in order to improve the system reliability while enabling feasible decoding. The diversity-multiplexing gain tradeoff (DMT) of the system together with sphere-decodability even with low number of antennas are used as the main design criteria, thus naturally establishing a DMT-complexity tradeoff. It is shown that the proposed protocol outperforms the simple time-division multiple access (TDMA) protocol, while still falling behind the optimal DMT.

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arXiv2013

Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

This work concentrates on the study of inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain trade-off is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain trade-off and point counting in Lie groups.

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arXiv2011

An algebraic look into MAC-DMT of lattice space-time codes

In this paper we are concentrating on the diversity-multiplexing gain trade-off (DMT) of some space-time lattice codes. First we give a DMT bound for lattice codes having restricted dimension. We then recover the well known results of the DMT of algebraic number field codes and the Alamouti code by using the union bound and see that these codes do achieve the previously mentioned bound. During our analysis interesting connections to the Dedekind's zeta-function and to the unit group of algebraic number fields are revealed. Finally we prove that both the number field codes and Alamouti code are in some sense optimal codes in the multiple access channel (MAC).

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arXiv2011

Diversity-multiplexing Gain Tradeoff: a Tool in Algebra?

Since the invention of space-time coding numerous algebraic methods have been applied in code design. In particular algebraic number theory and central simple algebras have been on the forefront of the research. In this paper we are turning the table and asking whether information theory can be used as a tool in algebra. We will first derive some corollaries from diversity-multiplexing gain (DMT) bounds by Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras. The authors do not claim that the algebraic results are new, but we do find that this interesting relation between algebra and information theory is quite surprising and worth pointing out.

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arXiv2009

On the Decay of the Determinants of Multiuser MIMO Lattice Codes

In a recent work, Coronel et al. initiated the study of the relation between the diversity-multiplexing tradeoff (DMT) performance of a multiuser multiple-input multiple-output (MU-MIMO) lattice code and the rate of the decay of the determinants of the code matrix as a function of the size of the signal constellation. In this note, we state a simple general upper bound on the decay function and study the promising code proposed by Badr and Belfiore in close detail. We derive a lower bound to its decay function based on a classical theorem due to Liouville. The resulting bound is applicable also to other codes with constructions based on algebraic number theory. Further, we study an example sequence of small determinants within the Badr-Belfiore code and derive a tighter upper bound to its decay function. The upper bound has certain conjectural asymptotic uncertainties, whence we also list the exact bound for several finite data rates.

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arXiv2006

Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) block code. This tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T>= n_t+n_r-1 where T is the number of time slots over which coding takes place and n_t,n_r are the number of transmit and receive antennas respectively. For T < n_t+n_r-1, only upper and lower bounds on the D-MG tradeoff are available. In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA) based ST codes having the non-vanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. For the rectangular, T > n_t case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all T>= n_t is the same as that previously known to hold for T >= n_t + n_r -1.

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