Generalized Sphere-Packing Bound for Subblock-Constrained Codes

We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem. In particular, we study binary constant subblock-composition codes (CSCCs), characterized by the property that the number of ones in each subblock is constant, and binary subblock energy-constrained codes (SECCs), characterized by the property that the number of ones in each subblock exceeds a certain threshold. For CSCCs, we show that the optimization problem is equivalent to finding the minimum of $N$ variables, where $N$ is independent of the number of subblocks. We then provide closed-form solutions for the generalized sphere-packing bounds for single- and double-error correcting CSCCs. For SECCs, we provide closed-form solutions for the generalized sphere-packing bounds for single errors in certain special cases. We also obtain improved bounds on the optimal asymptotic rate for CSCCs and SECCs, and provide numerical examples to highlight the improvement

On the Throughput of Channels that Wear Out

This work investigates the fundamental limits of communication over a noisy discrete memoryless channel that wears out, in the sense of signal-dependent catastrophic failure. In particular, we consider a channel that starts as a memoryless binary-input channel and when the number of transmitted ones causes a sufficient amount of damage, the channel ceases to convey signals. Constant composition codes are adopted to obtain an achievability bound and the left-concave right-convex inequality is then refined to obtain a converse bound on the log-volume throughput for channels that wear out. Since infinite blocklength codes will always wear out the channel for any finite threshold of failure and therefore cannot convey information at positive rates, we analyze the performance of finite blocklength codes to determine the maximum expected transmission volume at a given level of average error probability. We show that this maximization problem has a recursive form and can be solved by dynamic programming. Numerical results demonstrate that a sequence of block codes is preferred to a single block code for streaming sources.Comment: 23 pages, 1 table, 11 figures, submitted to IEEE Transactions on Communication

PERFORMANCE LIMITS FOR ENERGY-CONSTRAINED COMMUNICATION SYSTEMS

Ph.DDOCTOR OF PHILOSOPH

Estimating the Sizes of Binary Error-Correcting Constrained Codes

In this paper, we study binary constrained codes that are resilient to bit-flip errors and erasures. In our first approach, we compute the sizes of constrained subcodes of linear codes. Since there exist well-known linear codes that achieve vanishing probabilities of error over the binary symmetric channel (which causes bit-flip errors) and the binary erasure channel, constrained subcodes of such linear codes are also resilient to random bit-flip errors and erasures. We employ a simple identity from the Fourier analysis of Boolean functions, which transforms the problem of counting constrained codewords of linear codes to a question about the structure of the dual code. We illustrate the utility of our method in providing explicit values or efficient algorithms for our counting problem, by showing that the Fourier transform of the indicator function of the constraint is computable, for different constraints. Our second approach is to obtain good upper bounds, using an extension of Delsarte's linear program (LP), on the largest sizes of constrained codes that can correct a fixed number of combinatorial errors or erasures. We observe that the numerical values of our LP-based upper bounds beat the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).Comment: 51 pages, 2 figures, 9 tables, to be submitted to the IEEE Journal on Selected Areas in Information Theor

The Sphere Packing Bound for DSPCs with Feedback a la Augustin

Establishing the sphere packing bound for block codes on the discrete stationary product channels with feedback ---which are commonly called the discrete memoryless channels with feedback--- was considered to be an open problem until recently, notwithstanding the proof sketch provided by Augustin in 1978. A complete proof following Augustin's proof sketch is presented, to demonstrate its adequacy and to draw attention to two novel ideas it employs. These novel ideas (i.e., the Augustin's averaging and the use of subblocks) are likely to be applicable in other communication problems for establishing impossibility results.Comment: 12 pages, 2 figure

The Sphere Packing Bound via Augustin's Method

A sphere packing bound (SPB) with a prefactor that is polynomial in the block length $n$ is established for codes on a length $n$ product channel $W_{[1,n]}$ assuming that the maximum order $1/2$ Renyi capacity among the component channels, i.e. $\max_{t\in[1,n]} C_{1/2,W_{t}}$, is $\mathit{O}(\ln n)$. The reliability function of the discrete stationary product channels with feedback is bounded from above by the sphere packing exponent. Both results are proved by first establishing a non-asymptotic SPB. The latter result continues to hold under a milder stationarity hypothesis.Comment: 30 pages. An error in the statement of Lemma 2 is corrected. The change is inconsequential for the rest of the pape

Modules for Experiments in Stellar Astrophysics (MESA): Giant Planets, Oscillations, Rotation, and Massive Stars

We substantially update the capabilities of the open source software package Modules for Experiments in Stellar Astrophysics (MESA), and its one-dimensional stellar evolution module, MESA Star. Improvements in MESA Star's ability to model the evolution of giant planets now extends its applicability down to masses as low as one-tenth that of Jupiter. The dramatic improvement in asteroseismology enabled by the space-based Kepler and CoRoT missions motivates our full coupling of the ADIPLS adiabatic pulsation code with MESA Star. This also motivates a numerical recasting of the Ledoux criterion that is more easily implemented when many nuclei are present at non-negligible abundances. This impacts the way in which MESA Star calculates semi-convective and thermohaline mixing. We exhibit the evolution of 3-8 Msun stars through the end of core He burning, the onset of He thermal pulses, and arrival on the white dwarf cooling sequence. We implement diffusion of angular momentum and chemical abundances that enable calculations of rotating-star models, which we compare thoroughly with earlier work. We introduce a new treatment of radiation-dominated envelopes that allows the uninterrupted evolution of massive stars to core collapse. This enables the generation of new sets of supernovae, long gamma-ray burst, and pair-instability progenitor models. We substantially modify the way in which MESA Star solves the fully coupled stellar structure and composition equations, and we show how this has improved MESA's performance scaling on multi-core processors. Updates to the modules for equation of state, opacity, nuclear reaction rates, and atmospheric boundary conditions are also provided. We describe the MESA Software Development Kit (SDK) that packages all the required components needed to form a unified and maintained build environment for MESA. [Abridged]Comment: Accepted for publication in The ApJ Supplement Series. Extra informations required to reproduce the calculations in this paper are available at http://mesastar.org/results/mesa

Channel polarization: A method for constructing capacity-achieving codes

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity I(W) is the highest rate achievable subject to using the input letters of the channel equiprobably and equals the capacity C(W) if the channel has certain symmetry properties. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a different set of N binary-input channels such that the capacities of the latter set, except for a negligible fraction of them, are either near 1 or near 0. This second set of N channels are well-conditioned for channel coding: one need only send data at full rate through channels with capacity near 1 and at 0 rate through the others. The main coding theorem about polar coding states that, given any B-DMC W with I(W) > 0 and any fixed 0 < δ < I(W), there exist finite constants n1 (W, δ) and c(W, δ) such that for all n ≥ n1, there exist polar codes with block length N = 2n, rate R > I(W)-δ, and probability of block decoding error Pe ≤ cN-1/4. The codes with this performance can be encoded and decoded within complexity O(N log N). © 2008 IEEE