Abstract-The information rate of finite-state source/channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence, followed by a forward sum-product recursion on the joint source/channel trellis. This method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite-state approximations. Further upper and lower bounds can be computed by reduced-state methods.

Abstract-Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds.Because of the tight algebraic connection between QC codes and convolutional codes, we can state similar results for the free Hamming distance of convolutional codes. In fact, some QC code statements are established by first proving the corresponding convolutional code statements and then using a result by Tanner that says that the minimum Hamming distance of a QC code is upper bounded by the free Hamming distance of the convolutional code that is obtained by "unwrapping" the QC code.

SUMMARYWe consider linear-programming (LP) decoding of low-density parity-check (LDPC) codes. While one can use any general-purpose LP solver to solve the LP that appears in the decoding problem, we argue in this paper that the LP at hand is equipped with a lot of structure that can be exploited. Towards this goal, we study the dual LP and show how coordinate-ascent methods lead to very simple update rules that are tightly connected to the min-sum algorithm. Moreover, replacing minima in the formula of the dual LP with softminima one obtains update rules that are tightly connected to the sum-product algorithm. This shows that LP solvers with complexity similar to the min-sum algorithm and the sum-product algorithm are feasible.

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We present a design study for a nano-scale crossbar memory system that uses memristors with symmetrical but highly nonlinear current-voltage characteristics as memory elements. The memory is non-volatile since the memristors retain their state when un-powered. In order to address the nano-wires that make up this nano-scale crossbar, we use two coded demultiplexers implemented using mixed-scale crossbars (in which CMOS-wires cross nano-wires and in which the crosspoint junctions have one-time configurable memristors). This memory system does not utilize the kind of devices (diodes or transistors) that are normally used to isolate the memory cell being written to and read from in conventional memories. Instead, special techniques are introduced to perform the writing and the reading operation reliably by taking advantage of the nonlinearity of the type of memristors used. After discussing both writing and reading strategies for our memory system in general, we focus on a 64 x 64 memory array and present simulation results that show the feasibility of these writing and reading procedures. Besides simulating the case where all device parameters assume exactly their nominal value, we also simulate the much more realistic case where the device parameters stray around their nominal value: we observe a degradation in margins, but writing and reading is still feasible. These simulation results are based on a device model for memristors derived from measurements of fabricated devices in nano-scale crossbars using Pt and Ti nano-wires and using oxygen-depleted TiO(2) as the switching material.

We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph.Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences.The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call symbolwise graph-cover decoding, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help towards a better understanding of message-passing iterative decoding, namely blockwise graphcover decoding links max-product (min-sum) algorithm decoding with linear programming decoding, and symbolwise graph-cover decoding links sum-product algorithm decoding with Bethe free energy function minimization at temperature one.In contrast to the Gibbs entropy function, which is a concave function, the Bethe entropy function is in general not concave everywhere. In particular, we show that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain.Index Terms-Bethe approximation, Bethe entropy, Bethe partition function, graph cover, graph-cover decoding, messagepassing algorithm, method of types, linear programming decoding, pseudo-marginal vector, sum-product algorithm.

Some twenty years ago Margulis [l] proposed an algebraic construction of LDPC codes. In this paper we mimick the construction of Margulis and we describe a method for algebraically constructing regular and irregular LDPC codes.

Abstract-Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of timeinvariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework.Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the -mostly moderate -decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.