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Improved Constructions of Skew-Tolerant Gray Codes

Gabriel Sac Himelfarb, Moshe Schwartz

TL;DR

The first construction of asymptotically non-vanishing skew-tolerant Gray codes is presented, offering an exponential improvement over the known construction and linear-time encoding and decoding algorithms for the codes are provided.

Abstract

We study skew-tolerant Gray codes, which are Gray codes in which changes in consecutive codewords occur in adjacent positions. We present the first construction of asymptotically non-vanishing skew-tolerant Gray codes, offering an exponential improvement over the known construction. We also provide linear-time encoding and decoding algorithms for our codes. Finally, we extend the definition to non-binary alphabets, and provide constructions of complete $m$-ary skew-tolerant Gray codes for every base $m\geq 3$.

Improved Constructions of Skew-Tolerant Gray Codes

TL;DR

The first construction of asymptotically non-vanishing skew-tolerant Gray codes is presented, offering an exponential improvement over the known construction and linear-time encoding and decoding algorithms for the codes are provided.

Abstract

We study skew-tolerant Gray codes, which are Gray codes in which changes in consecutive codewords occur in adjacent positions. We present the first construction of asymptotically non-vanishing skew-tolerant Gray codes, offering an exponential improvement over the known construction. We also provide linear-time encoding and decoding algorithms for our codes. Finally, we extend the definition to non-binary alphabets, and provide constructions of complete -ary skew-tolerant Gray codes for every base .

Paper Structure

This paper contains 13 sections, 11 theorems, 56 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary skew-tolerant Gray codes
  4. The case $k=3$
  5. The case $k=2$
  6. The case $k=1$
  7. $m$-ary skew-tolerant Gray codes
  8. $m$-ary codes, $m\geqslant 5$
  9. Quaternary codes
  10. Ternary codes
  11. Conclusion
  12. Base case for Construction \ref{['const:skip1general']}, $N_0=6$, $L=4$, $R=1$
  13. Base case for Construction \ref{['const:skip1general']}, $N_0=7$, $L=3$, $R=3$

Key Result

Theorem 5

For every $n\geqslant 1$, $A_n$ from Construction const:skip3 is a complete $3$-SkTGC of length $n$.

Figures (1)

  • Figure 1: Reading heads mounted over a surface (a) without a skew, and (b) with a skew.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Example 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • Theorem 10
  • ...and 5 more