Skip-Sliding Window Codes

Constrained coding is used widely in digital communication and storage systems. In this article, 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 <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> of a sliding window, skip length <inline-formula> <tex-math notation="LaTeX">$J$ </tex-math></inline-formula>, and cost constraint <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> in each sliding window. Each valid codeword of length <inline-formula> <tex-math notation="LaTeX">$L + kJ$ </tex-math></inline-formula> is determined by <inline-formula> <tex-math notation="LaTeX">$k+1$ </tex-math></inline-formula> windows of length <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> where window <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> starts at <inline-formula> <tex-math notation="LaTeX">$(iJ + 1)$ </tex-math></inline-formula>th symbol for all non-negative integers <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$i \leq k$ </tex-math></inline-formula>; and the cost constraint <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> in each window must be satisfied. SSW coding constraints naturally arise in applications such as simultaneous energy and information transfer, and SSW codes are also potential candidates for visible light communications. 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 some useful observations are made, such as the fact that SSW codes provide greater capacity than certain related classes of constrained codes. Moreover, we provide noisy capacity bounds for SSW codes.