[Paper Review] Skip-Sliding Window Codes

Ting-Yi Wu, Anshoo Tandon+2 more|arXiv (Cornell University)|Nov 27, 2017

Algorithms and Data Compression19 references3 citations

TL;DR

This paper introduces skip-sliding window (SSW) codes, a generalized constrained coding scheme that interpolates between sliding window and subblock-energy constraints by enforcing a minimum Hamming weight $W$ in every $L$-length window starting at positions $iJ+1$ for $i = 0,1,...,k$. The authors develop two enumeration methods—based on modified de Bruijn graphs and a refined Goulden-Jackson cluster method—enabling efficient computation of noiseless capacity, which surprisingly exceeds that of subblock-energy codes under certain parameters, challenging the intuition that tighter constraints always reduce capacity.

ABSTRACT

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.

Motivation & Objective

• To formalize skip-sliding window (SSW) codes as a generalized constrained coding framework that unifies sliding window and subblock-energy constraints.
• To develop efficient enumeration techniques for counting valid SSW codewords under a minimum Hamming weight constraint $W$.
• To analyze the noiseless capacity of binary SSW codes and identify conditions under which they outperform existing constrained codes.
• To derive upper and lower bounds on noisy channel capacity over BSC and BEC for SSW codes.
• To explore applications in simultaneous information and energy transfer, genome sequence assembly, and task scheduling under load constraints.

Proposed method

• Proposes a mathematical model for SSW codes defined by parameters $L$ (window length), $J$ (skip length), and $W$ (minimum Hamming weight), where valid codewords must satisfy the weight constraint in $k+1$ non-overlapping windows starting at $iJ+1$ for $i = 0$ to $k$.
• Develops a first enumeration method using the adjacency matrix of a modified de Bruijn graph, with complexity $Oig{(}ig{(}\sum_{i=W}^{L}\binom{L}{i}\big{)}^2\big{)}$.
• Introduces a second enumeration method based on the Goulden-Jackson cluster method, applicable when $L = \ell J$, with complexity $O\big{(}\big{(}\sum_{i=0}^{W-1}\binom{L}{i}\big{)}^2\big{)}$.
• Refines both methods for $L = \ell J$, reducing complexity to $O\big{(}\min\{J+1,W+1\}^{\ell-1}\big{)}$ and $O\big{(}\min\{J,W-1\}^{\ell-1}\big{)}$ respectively.
• Proves equivalence between the refined de Bruijn and Goulden-Jackson methods under the $L = \ell J$ condition.
• Derives noisy capacity bounds for SSW codes over binary symmetric (BSC) and binary erasure (BEC) channels using super-letter modeling and the Blahut-Arimoto algorithm for SEC code capacity.

Experimental results

Research questions

• RQ1 Can a generalized constrained coding scheme that relaxes the sliding window constraint by skipping some windows achieve higher noiseless capacity than traditional subblock-energy codes?
• RQ2 What efficient enumeration techniques can be used to compute the size of SSW codes under a minimum Hamming weight constraint?
• RQ3 Does the noiseless capacity of SSW codes exceed that of subblock-energy codes for certain parameter sets, defying the intuition that tighter constraints reduce capacity?
• RQ4 How do the noisy capacities of SSW codes compare to those of SEC codes over BSC and BEC?
• RQ5 Can the proposed SSW framework be applied to real-world problems such as simultaneous information and energy transfer, genome assembly, and task scheduling?

Key findings

• The noiseless capacity of SSW codes can exceed that of subblock-energy (SEC) codes for certain parameter sets, such as $(L,J,W) = (8,7,7)$, contradicting the expectation that looser constraints always yield higher capacity.
• For $L = \ell J$, the refined enumeration method based on the de Bruijn graph reduces complexity to $O\big{(}\min\{J+1,W+1\}^{\ell-1}\big{)}$, significantly improving scalability.
• The refined Goulden-Jackson method reduces complexity to $O\big{(}\min\{J,W-1\}^{\ell-1}\big{)}$, offering a more efficient alternative for specific parameter regimes.
• Numerical results show that the lower bound on noisy capacity using Mrs. Gerber’s Lemma is uniformly worse than the bound derived via super-letter modeling, with the latter being tighter for $p \to 0$ and $p \to 0.5$ in BSC.
• Over the BEC, the two lower bounds on capacity coincide and become tight as $\epsilon \to 0$ and $\epsilon \to 1$, indicating strong agreement in asymptotic regimes.
• The upper bound on noisy capacity for SSW codes is tighter than the noiseless capacity and also bounded by the SEC code capacity, confirming the validity of the derived bounds.