Fabian Lim

The Single-Uniprior Index-Coding Problem: The Single-Sender Case and the Multi-Sender Extension

Index coding studies multiterminal source-coding problems where a set of receivers are required to decode multiple (possibly different) messages from a common broadcast, and they each know some messages a priori. In this paper, at the receiver end, we consider a special setting where each receiver knows only one message a priori, and each message is known to only one receiver. At the broadcasting end, we consider a generalized setting where there could be multiple senders, and each sender knows a subset of the messages. The senders collaborate to transmit an index code. This paper looks at minimizing the number of total coded bits the senders are required to transmit. When there is only one sender, we propose a pruning algorithm to find a lower bound on the optimal (i.e., the shortest) index codelength, and show that it is achievable by linear index codes. When there are two or more senders, we propose an appending technique to be used in conjunction with the pruning technique to give a lower bound on the optimal index codelength; we also derive an upper bound based on cyclic codes. While the two bounds do not match in general, for the special case where no two distinct senders know any message in common, the bounds match, giving the optimal index codelength. The results are expressed in terms of strongly connected components in directed graphs that represent the index-coding problems.

Ordered Statistics Decoding of Linear Block Codes Over Intersymbol Interference Channels

In this paper, we utilize ordered statistics decoding (OSD) techniques for intersymbol interference (ISI) channels. To achieve good efficiency and storage complexity, a combination of a modified generalized Viterbi algorithm (GVA) and Battails algorithm is proposed to implement OSD for ISI channels. Simulation results showed significant performance improvement for the [128,64,22] eBCH code, and the [255,239,17] Reed-Solomon (RS) binary image, on the (1+D)2 perpendicular recording channel.

Code Automorphisms and Permutation Decoding of Certain Reed–Solomon Binary Images

We consider primitive Reed-Solomon (RS) codes over the field F2m of length n=2m-1. Building on Lacan s results for the case of binary extension fields, we show that the binary images of certain two-parity symbol RS [n, n-2, 3] code, have a code automorphism subgroup related to the general linear group GL(m, 2). For these codes, we obtain a code automorphism subgroup of order m! GL(m,2). An explicit algorithm is given to compute a code automorphism (if it exists), that sends a particular choice of m binary positions, into binary positions that correspond to a single symbol of the RS code. If one such automorphism exists for a particular choice of m binary symbol positions, we show that there are at least m! of them. Computationally efficient permutation decoders are designed for the two-parity symbol RS [n, n-2, 3] codes. Simulation results are shown for the additive white Gaussian noise (AWGN) channel. For the finite fields F23 and F24, we go on to derive subgroups of code automorphisms, belonging to binary images of certain RS codes that have three-parity symbols. A table of code automorphism subgroup orders, computed using the Groups, Algorithms, and Programming (GAP) software, is tabulated for the fields F23, F24 , and F25.

Linear programming upper bounds on permutation code sizes from coherent configurations related to the Kendall-tau distance metric

Recent interest on permutation rank modulation shows the Kendall-tau metric as an important distance metric. This note documents our first efforts to obtain upper bounds on optimal code sizes (for said metric) ala Delsartes approach. For the Hamming metric, Delsartes seminal work on powerful linear programming (LP) bounds have been extended to permutation codes, via association scheme theory. For the Kendall-tau metric, the same extension needs the more general theory of coherent configurations, whereby the optimal code size problem can be formulated as an extremely huge semidefinite programming (SDP) problem. Inspired by recent algebraic techniques for solving SDPs, we consider the dual problem, and propose an LP to search over a subset of dual feasible solutions. We obtain modest improvement over a recent Singleton bound due to Barg and Mazumdar. We regard this work as a starting point, towards fully exploiting the power of Delsartes method, which are known to give some of the best bounds in the context of binary codes.

The multi-sender multicast index coding

We focus on the following instance of an index coding problem, where a set of receivers are required to decode multiple messages, whilst each knows one of the messages a priori. In particular, here we consider a generalized setting where they are multiple senders, each sender only knows a subset of messages, and all senders are required to collectively transmit the index code. For a single sender, Ong and Ho (ICC, 2012) have established the optimal index codelength, where the lower bound was obtained using a pruning algorithm. In this paper, the pruning algorithm is simplified, and used in conjunction with an appending technique to give a lower bound to the multi-sender case. An upper bound is derived based on network coding. While the two bounds do not match in general, for the special case where no two senders know any message bit in common, the bounds match, giving the optimal index codelength. The results are derived based on graph theory, and are expressed in terms of strongly connected components.

Permutation decoding the binary images of certain double-parity reed-solomon codes

We introduce two permutation decoder designs for the binary images of double-parity [n, n − 2, 3] Reed-Solomon (RS) codes over binary extension fields F2m. The codes considered are limited to have zeros {1, α}, where α is any primitive element in F2m. We show that there exists a large set of m binary symbol errors that may be corrected via permutation decoding. The permutation decoders are shown to achieve near maximum-likelihood decoder performance, while only utilizing simple ideas borrowed from well-known reliability-based decoding algorithms.

Notes on the automorphism groups of Reed Solomon binary images

In this paper, a new proof of the work by Lacan et. al is obtained. This approach led to newly discovered connections between the automorphism group of the Reed Solomon (RS) binary image, and elementary group theory. This new development facilitated proving the existence and constructing permutations that have not been previously reported in the literature. Simple special cases are then considered in this work.

Reliability Distributions of Truncated Max-Log-MAP (MLM) Detectors Applied to Binary ISI Channels

The max-log-MAP (MLM) receiver is an approximated version of the well-known Bahl-Cocke-Jelinek-Raviv algorithm. The MLM algorithm is attractive due to its implementation simplicity. In practice, sliding-window implementations are preferred, whereby truncated signaling neighborhoods (around each transmission time instant) are considered. In this paper, we consider binary signaling sliding-window MLM receivers, where the MLM detector is truncated to a length-m signaling neighborhood. Here, truncation is used to ease the burden of analysis. For any number n of chosen times instants, we derive exact expressions for both 1) the joint distribution of the MLM symbol reliabilities, and 2) the joint probability of the erroneous MLM symbol detections. We show that the obtained expressions can be efficiently evaluated using Monte-Carlo techniques. The most computationally expensive operation (in each Monte-Carlo trial) is an eigenvalue decomposition of a size 2mn×2mn matrix. The proposed method handles various scenarios such as correlated noise distributions, modulation coding, etc.

List Decoding Techniques for Intersymbol Interference Channels Using Ordered Statistics

In this paper, we present a generalization of the ordered statistics decoding (OSD) techniques for the class of intersymbol interference (ISI) channels, and show decoding results for the extended Bose-Chaudhuri-Hocquenghem (eBCH) [128,64,22] code and the [255,239,17] Reed-Solomon (RS) binary image, over the PR2 partial response channel. Using the generalized OSD technique, we go on to generalize the Box-andMatch Algorithm (BMA) to the class of ISI channels. The BMA is an enhancement of OSD, and prior work has shown it to provide significant performance gain over OSD for memoryless additive white Gaussian noise (AWGN) channels. We present decoding results of the BMA for ISI channels, for the same eBCH and RS (binary image) codes, and PR2 channel. Our results show that the BMA (generalized for ISI channels) is superior to the OSD in terms of its performance/complexity trade-off. More specifically, the BMA may be tuned such that both algorithms have similar complexity, whereby the BMA still outperforms the OSD by a significant margin.

Computing reliability distributions of windowed max-log-map (MLM) detectors : ISI channels

In this paper, we consider sliding-window max-log-map (MLM) receivers, where for any integer m, the MLM detector is truncated to a length-m signaling neighborhood. For any number n of chosen time instants, we provide exact expressions for both i) the joint distribution of the MLM symbol reliabilities, and ii) the joint probability of the erroneous MLM symbol detections. The obtained expressions can be efficiently evaluated using Monte-Carlo techniques. Comparisons performed with empirical distributions reveal good match. Dynamic programming techniques are applied to simplify the procedures.