Chromatic Codes for Latency Optimal Geo-Distributed Storage

The problem of compression in standard information theory consists of assigning codes as short as possible to numbers. Here we consider the problem of optimal coding – under an arbitrary coding scheme – and show that it predicts Zipf’s law of abbreviation, namely a tendency in natural languages for more frequent words to be shorter. We apply this result to investigate optimal coding also under so-called non-singular coding, a scheme where unique segmentation is not warranted but codes stand for a distinct number. Optimal non-singular coding predicts that the length of a word should grow approximately as the logarithm of its frequency rank, which is again consistent with Zipf’s law of abbreviation. Optimal non-singular coding in combination with the maximum entropy principle also predicts Zipf’s rank-frequency distribution. Furthermore, our findings on optimal non-singular coding challenge common beliefs about random typing. It turns out that random typing is in fact an optimal coding process, in stark contrast with the common assumption that it is detached from cost cutting considerations. Finally, we discuss the implications of optimal coding for the construction of a compact theory of Zipfian laws more generally as well as other linguistic laws.

In this paper, we consider the problem of differential space-time (ST) block coding in the context of impulse-radio ultra-wideband (IR-UWB) communications with two transmit antennas and Pulse Position Modulation (PPM). We propose a novel family of codes where the information is encoded differentially through the number of transmitted pulses, the positions of these pulses and the locations of the non-idle symbol durations within each ST block. In order to maintain a low complexity of the UWB transmitter, unipolar transmissions are ensured by avoiding all forms of amplitude-domain encoding. The proposed scheme provides a full scalability capability where depending on the value taken by the scale parameter Θ, different levels of compromise between data rates and error rates (along with the decoding complexity) can be achieved. Finally, the proposed scheme lends itself to optimal detection with cross-correlation receivers that can be implemented in the analog domain in the absence of all forms of channel state information at the transmitter and receiver sides.

Index coding is an elegant and powerful idea for minimal usage of broadcast channels that investigates the fundamental limits and optimal coding schemes in information flow networks such as wireless networks, data storage networks, and content delivery networks. Minimal usage of broadcast channels is the main requirement of efficient information broadcasting in information flow networks which is the ultimate goal for successful communication. Finding the optimal index code with extreme rate for the class of index coding problems is the fundamental question in index coding. In this work, we present a new class of index coding problems with the size of maximum acyclic induced subgraph = 3. We refer to this class of index coding problems as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problems. Our goal is to characterize the extreme rate (i.e., the optimal index code length for extreme rate) for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problems. We show that the optimal linear index code of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$class \mathbb {III}$ </tex-math></inline-formula> index coding problem over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{2}$ </tex-math></inline-formula> achieves the extreme rate on the broadcast rate. To present this new result, we use the results of index coding network coding duality. In contrast with this new result, we also present that there exists maximum acyclic induced subgraph size 3 index coding problems with the optimal broadcast rate strictly greater than the extreme rate. To present this contrast, we use the alignment graph and the conflict hyper graph based approach.

This paper investigates the structures and properties of one-Lee weight codes and two-Lee weight projective codes over ℤ4. The authors first give the Pless identities on the Lee weight of linear codes over ℤ4. Then the authors study the necessary conditions for linear codes to have one-Lee weight and two-Lee projective weight respectively, the construction methods of one-Lee weight and two-Lee weight projective codes over ℤ4 are also given. Finally, the authors recall the weight-preserving Gray map from (ℤ 4 , Lee weight) to ( $\mathbb{F}_2^{2n} $ , Hamming weight), and produce a family of binary optimal oneweight linear codes and a family of optimal binary two-weight projective linear codes, which reach the Plotkin bound and the Griesmer bound.

The design of space-time codes for frequency flat, spatially correlated MIMO fading channels is considered. The focus of the paper is on the class of space-time block codes known as linear dispersion (LD) codes, introduced by Hassibi and Hochwald. The LD codes are optimized with respect to the mutual information between the inputs to the space-time encoder and the output of the channel. The use of the mutual information as both a design criterion and a performance measure is justified by allowing soft decisions at the output of the space-time decoder. A spatial Fourier (virtual) representation of the channel is exploited to allow for the analysis of MIMO channels with quite general fading statistics. Conditions, known as generalized orthogonality conditions (GOC), are derived for an LD code to achieve an upper bound on the mutual information, with the understanding that LD codes that achieve the upper bound, if they exist, are optimal. Explicit code constructions and properties of the optimal power allocation schemes are also derived. In particular, it is shown that optimal LD codes correspond to beamforming to a single virtual transmit angle at low SNR, and a necessary and sufficient condition for beamforming to be optimal is provided. Finally, numerical results are provided to illustrate the optimal code design for two examples of sparse scattering environments. The performance of the optimal LD codes for these scattering environments is compared with that of LD codes designed assuming the i.i.d. Rayleigh (rich scattering) model, and it is shown that the optimal LD codes perform significantly better. The optimal LD codes are also compared to beamforming LD codes and it is shown that beamforming is nearly optimal over a range of SNR's of interest.

&lt;p style='text-indent:20px;'&gt;Recently, some infinite families of optimal binary linear codes are constructed from simplicial complexes. Afterwards, the construction method was extended to using arbitrary posets. In this paper, based on a generic construction of linear codes, we obtain four classes of optimal binary linear codes by using the posets of two chains. Two of them induce Griesmer codes which are not equivalent to the linear codes constructed by Belov. Those codes are exploited to construct secret sharing schemes in cryptography as well.&lt;/p&gt;

Route optimization support in mobile networks is one of the most important concerns in Next Generation Networks (NGNs). NGNs require an efficient route optimization scheme, optimized specially to support nesting of mobile networks. We propose an Extended Mobile IPv6 route optimization (EMIP) scheme to enhance the performance of nested mobile networks. EMIP scheme is based on MIPv6 route optimization and the root Mobile Router (MR) performs all the route optimization tasks on behalf of all active Mobile Network Nodes (MNNs). Thus, the network movement remains transparent to sub MRs and MNNs. It modifies only MRs and MNNs leaving other entities untouched and is more efficient than the Network Mobility Basic Support protocol (NEMO BS). We carried out extensive simulation study to show that EMIP performs better than NEMO BS and Reverse Routing Header (RRH, a well established route optimization protocol) with respect to TCP throughput and round trip time. EMIP effectively reduces unnecessary large travelling route and overhead, thereby enhances performance. It also achieves route optimization between the mobile network nodes inside nested mobile network.

&lt;p style='text-indent:20px;'&gt;Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; prime or composite, cyclotomy on a residue class ring &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ {\mathbb{Z}}_{v} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [&lt;xref ref-type="bibr" rid="b40"&gt;40&lt;/xref&gt;], we introduce a generalized cyclotomy of order &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ e $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on the ring &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, where &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ q_i $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ q_j $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ i\neq j $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) may not be co-prime, which includes classical cyclotomy as a special case. Here, &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ q_1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ q_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \cdots $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ q_k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are powers of primes with an integer &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ e|(q_i-1) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for any &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ 1\leq i\leq k $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.&lt;/p&gt;

We study the problem of finding zero-error instantaneous codes for the Slepian-Wolf configuration. By a zero-error instantaneous code we mean two encoder maps f/sub 1/:/spl chi//spl rarr/{0,1}*, f/sub 2/:Y/spl rarr/{0,1}* and a decoding algorithm which, for any pair of encoder outputs f/sub 1/(x/sub 1/)f/sub 1/(x/sub 2/)f/sub 1/(x/sub 3/)... and f/sub 2/(y/sub 1/)f/sub 2/(y/sub 2/)f/sub 2/(y/sub 3/)..., can correctly determine x/sub 1/ and y/sub 1/ by reading only the first length (f/sub 1/(x/sub 1/)) bits of the X encoder output and the first length (f/sub 2/(y/sub 1/)) bits of the Y encoder output. For |/spl chi/|=2, we find a necessary and sufficient for the existence of a zero-error instantaneous code with a given set of codeword lengths for Y. Using this condition, we derive an upper bound to the minimum expected codeword length for Y and construct a simple example showing that the coding scheme proposed by Kh, Jabri and Al-Issa (1997) is not optimal in general, and more surprisingly, the optimal code may violate the Morse condition that the more probable of two symbols never has the longer codeword. For |/spl chi/|=3, we find a necessary but not sufficient for the existence of a zero-error instantaneous code with a given set of codeword lengths for Y. Moreover, for |/spl chi/|/spl ges/3, the existence of such a code is shown to be related to a rectangle packing problem.

There is a large literature devoted to the problem of finding an optimal (min-cost) prefix-free code with an unequal letter-cost encoding alphabet of size. While there is no known polynomial time algorithm for optimally solving it, there are many good heuristics that all provide additive errors to optimal. The additive error in these algorithms usually depends linearly upon the size of the largest encoding letter. This paper was motivated by the problem of finding optimal codes when the encoding alphabet is infinite. Because the largest letter cost is infinite, the previous analyses could give infinite error bounds. We provide a new algorithm that works with infinite encoding alphabets. When restricted to the finite alphabet case, our algorithm often provides better error bounds than the best previous ones known.

Malicious clients in distributed storage networks. The presentation will be split up into two parts. In the initial part, we’re going to talk about distributed storage networks that use secret-sharing, and in the second part I’m going to introduce malicious clients, their possible attack vectors and how to protect against those attack vectors.

The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance Algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space Rn. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.

In distributed storage systems, storage nodes intermittently go offline for numerous reasons. On coming back online, nodes need to update their contents to reflect any modifications to the data in the interim. In this paper, we consider a setting where no information regarding modified data needs to be logged in the system. In such a setting, a 'stale' node needs to update its contents by downloading data from already updated nodes, while neither the stale node nor the updated nodes have any knowledge as to which data symbols are modified and what their value is. We investigate the fundamental limits on the amount of communication necessary for such an "oblivious" update process. We first present a generic lower bound on the amount of communication that is necessary under any storage code with a linear encoding (while allowing non-linear update protocols). This lower bound is derived under a set of extremely weak conditions, giving all updated nodes access to the entire modified data and the stale node access to the entire stale data as side information. We then present codes and update algorithms that are optimal in that they meet this lower bound. Next, we present a lower bound for an important subclass of codes, that of linear Maximum-Distance-Separable (MDS) codes. We then present an MDS code construction and an associated update algorithm that meets this lower bound. These results thus establish the capacity of oblivious updates in terms of the communication requirements under these settings.

We study the problem of lossless joint sourcechannel coding (JSCC) in the finite block length regime from an unequal message protection (UMP) perspective. We demonstrate that the problem of lossless JSCC can be cast in terms of UMP codes previously studied. We show that an optimal JSCC can be constructed from a matched UMP code. We further derive a finite block length bound that characterizes the performance of a JSCC constructed from a UMP code not perfectly matched to the source. This bound is evaluated for a binary memoryless source transmitted over a binary symmetric channel. Two-class schemes previously studied in literature are compared with the proposed scheme. Empirically the JSCCs based on UMP codes approach the performance of the optimal matched code quite fast in number of classes used.

The problem of sequential source coding is to minimize the average entropy rate subject to a constraint on the average distortion and a constraint on codewords. This is cast as an optimization problem on an appropriate convex set of probability measures. Existence and properties of optimal sequential codes are explored. A sequential rate distortion theorem is proved and a construction given to show that in general, a causality gap exists.