PERFORMANCE LIMITS FOR ENERGY-CONSTRAINED COMMUNICATION SYSTEMS

Ph.DDOCTOR OF PHILOSOPH

Skip-Sliding Window Codes

Constrained coding is used widely in digital communication and storage systems. In this paper, we study a generalized sliding window constraint called the skip-sliding window. A skip-sliding window (SSW) code is defined in terms of the length $L$ of a sliding window, skip length $J$, and cost constraint $E$ in each sliding window. Each valid codeword of length $L + kJ$ is determined by $k+1$ windows of length $L$ where window $i$ starts at $(iJ + 1)$th symbol for all non-negative integers $i$ such that $i \leq k$; and the cost constraint $E$ in each window must be satisfied. In this work, two methods are given to enumerate the size of SSW codes and further refinements are made to reduce the enumeration complexity. Using the proposed enumeration methods, the noiseless capacity of binary SSW codes is determined and observations such as greater capacity than other classes of codes are made. Moreover, some noisy capacity bounds are given. SSW coding constraints arise in various applications including simultaneous energy and information transfer.Comment: 28 pages, 11 figure

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

Galois Theory and Solvable Equations of Prime Degree

In this article we review classical and modern Galois theory with historical evolution and prove a criterion of Galois for solvability of an irreducible separable polynomial of prime degree over an arbitrary field k and give many illustrative examples