The maximum length of circuit codes with long bit runs and a new characterization theorem

Abstract

We study circuit codes with long bit runs (sequences of distinct transitions) and derive a formula for the maximum length for an infinite class of symmetric circuit codes with long bit runs. This formula also results in an improved lower bound on the maximum length for an infinite class of circuit codes without restrictions on symmetry or bit run length. We also present a new characterization of circuit codes of spread k based on a theorem of Deimer.

On the Asymptotic Order of Circuit Codes

“…Our notation follows [2], throughout this note we treat vertices of I(d) and their corresponding binary vectors of length d interchangeably. A (d, k) circuit code C = (x 1 , .…”

The Maximum Length and Isomorphism of Circuit Codes with Long Bit Runs

“…) and C is symmetric}. By assuming both symmetry and the existence of a long bit run, [5] was able to extend the family of (d, k) pairs for which a formula for the maximum length was known, proving the following. Theorem 1.2 ([5] Theorem 3.9).…”

“…Circuit codes have been extensively studied in the case k = 2, where they are called 'snakes in the box' or 'coils in the box' [8,1,27,25,29,10,18,19,21,2]. The general case where k ∈ N is surveyed in chapter 17 of [11] and studied in detail in [24,16,9,23,22,12,7,14,4,5]. Vertices of I(d) can be considered as vectors in {0, 1} d .…”

“…When k and l have opposite parity this lower bound simplifies to d ≥ 1 2 (3k + l + 1). Generalizable constructions exist where this inequality is tight [9,5], and we say "C has a long bit run" to denote such instances. In addition, ( [24] Theorem 1) also established that for any…”

“…In addition, he proved that when k is odd, any two maximum length (⌊ 3k 2 ⌋ + 2, k) circuit codes are isomorphic ([9] Theorem 4). Recently Byrnes [5] observed that in each of the preceding cases where a formula for K(d, k) was known, it was possible to construct a maximum length circuit code C that possessed a long bit run and which was symmetric, i.e.…”