12 research outputs found
A duality for nonabelian group codes
In 1962, Jesse MacWilliams published a set of formulas for linear and abelian
group codes that among other applications, were incredibly valuable in the
study of self-dual codes. Now called the MacWilliams Identities, her results
relate the weight enumerator and complete weight enumerator of a code to those
of its dual code. A similar set of MacWilliams identities has been proven to
exist for many other types of codes. In 2013, Dougherty, Sol\'{e}, and Kim
published a list of fundamental open questions in coding theory. Among them,
Open Question 4.3: "Is there a duality and MacWilliams formula for codes over
non-Abelian groups?" In this paper, we propose a duality for nonabelian group
codes in terms of the irreducible representations of the group. We show that
there is a Greene's Theorem and MacWilliams Identities which hold for this
notion of duality. When the group is abelian, our results are equivalent to
existing formulas in the literature.Comment: 12 page
Media-Based MIMO: A New Frontier in Wireless Communications
The idea of Media-based Modulation (MBM), is based on embedding information
in the variations of the transmission media (channel state). This is in
contrast to legacy wireless systems where data is embedded in a Radio Frequency
(RF) source prior to the transmit antenna. MBM offers several advantages vs.
legacy systems, including "additivity of information over multiple receive
antennas", and "inherent diversity over a static fading channel". MBM is
particularly suitable for transmitting high data rates using a single transmit
and multiple receive antennas (Single Input-Multiple Output Media-Based
Modulation, or SIMO-MBM). However, complexity issues limit the amount of data
that can be embedded in the channel state using a single transmit unit. To
address this shortcoming, the current article introduces the idea of Layered
Multiple Input-Multiple Output Media-Based Modulation (LMIMO-MBM). Relying on a
layered structure, LMIMO-MBM can significantly reduce both hardware and
algorithmic complexities, as well as the training overhead, vs. SIMO-MBM.
Simulation results show excellent performance in terms of Symbol Error Rate
(SER) vs. Signal-to-Noise Ratio (SNR). For example, a LMIMO-MBM is
capable of transmitting bits of information per (complex) channel-use,
with SER at dB (or SER
at dB). This performance is achieved using a single transmission
and without adding any redundancy for Forward-Error-Correction (FEC). This
means, in addition to its excellent SER vs. energy/rate performance, MBM
relaxes the need for complex FEC structures, and thereby minimizes the
transmission delay. Overall, LMIMO-MBM provides a promising alternative to MIMO
and Massive MIMO for the realization of 5G wireless networks.Comment: 26 pages, 11 figures, additional examples are given to further
explain the idea of Media-Based Modulation. Capacity figure adde
On Grid Codes
If is finite alphabet for , the Manhattan distance is
defined in . A grid code is introduced as a subset of
. Alternative versions of the Hamming and
Gilbert-Varshamov bounds are presented for grid codes. If is a cyclic
group for , some bounds for the minimum Manhattan distance of codes
that are cyclic subgroups of are determined in terms of
their minimum Hamming and Lee distances. Examples illustrating the main results
are provided
Constructing Linear Encoders with Good Spectra
Linear encoders with good joint spectra are suitable candidates for optimal
lossless joint source-channel coding (JSCC), where the joint spectrum is a
variant of the input-output complete weight distribution and is considered good
if it is close to the average joint spectrum of all linear encoders (of the
same coding rate). In spite of their existence, little is known on how to
construct such encoders in practice. This paper is devoted to their
construction. In particular, two families of linear encoders are presented and
proved to have good joint spectra. The first family is derived from Gabidulin
codes, a class of maximum-rank-distance codes. The second family is constructed
using a serial concatenation of an encoder of a low-density parity-check code
(as outer encoder) with a low-density generator matrix encoder (as inner
encoder). In addition, criteria for good linear encoders are defined for three
coding applications: lossless source coding, channel coding, and lossless JSCC.
In the framework of the code-spectrum approach, these three scenarios
correspond to the problems of constructing linear encoders with good kernel
spectra, good image spectra, and good joint spectra, respectively. Good joint
spectra imply both good kernel spectra and good image spectra, and for every
linear encoder having a good kernel (resp., image) spectrum, it is proved that
there exists a linear encoder not only with the same kernel (resp., image) but
also with a good joint spectrum. Thus a good joint spectrum is the most
important feature of a linear encoder.Comment: v5.5.5, no. 201408271350, 40 pages, 3 figures, extended version of
the paper to be published in IEEE Transactions on Information Theor
Algebraic Structures for Multi-Terminal Communication Systems.
We study a distributed source coding problem with multiple encoders, a central decoder and a joint distortion criterion. The encoders do not communicate with each other. The encoders observe correlated sources which they quantize and communicate noiselessly to a central decoder which is interested in minimizing a joint distortion criterion that depends on the sources and the reconstruction. We are interested in characterizing an inner bound to the optimal rate-distortion region. We first consider a special case where the sources are jointly Gaussian and the decoder wants to reconstruct a linear function of the sources under mean square error distortion. We demonstrate a coding scheme involving nested lattice codes that reconstructs the linear function by encoding in such a fashion that the decoder is able to reconstruct the function directly. For certain source distributions, this approach yields a larger rate-distortion region compared to when the decoder reconstructs lossy versions of the sources first and then estimates the function from them. We then
extend this approach to the case of reconstructing a linear function of an arbitrary
number of jointly Gaussian sources. Next, we consider the general distributed source coding problem with discrete sources. This formulation includes as a special case many famous distributed source coding problems. We present a new achievable rate-distortion region for this problem based on “good” structured nested random codes built over abelian groups. We demonstrate rate gains for this problem over traditional coding schemes using unstructured random codes. For certain sources and distortion functions, the new rate region is strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for the problem till now. Further, there is no known way of achieving these rate gains without exploiting the structure of the coding scheme. Achievable performance limits for single-user source coding using abelian group codes are also obtained as corollaries of the main coding theorem. Our results also imply that nested linear codes achieve the Shannon rate-distortion bound in the single-user setting. Finally, we conclude by outlining some future research directions.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/75917/1/dineshk_1.pd