On the maximum length of coil-in-the-box codes in dimension 8

New Bounds for the Snake-in-the-Box Problem

“…-For n = 7, Potter, Robinson, Miller and Kochut [16] used a genetic algorithm to find a longest snake and Kochut [11] used exhaustive search to find a longest coil. -For n = 8, Östergård and Pettersson used canonical augmentation to find a longest snake [14] and a longest coil [15].…”

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

“…Unlike the case with the upper bounds of Propositions 3.2 and 3.3, the upper bound of 4k + 2l can be achieved. This is shown in the following example which is based on a canonical augmentation approach similar to [16].…”

“…This is not true for K(⌊ 3k 2 ⌋ + 3, k) codes with k odd. Consider the (16,9) code C defined by the transition sequence T of Example 1, it has length 44 = K (16,9) by Theorem 3.4 (iii). Another (16,9) code of length 44 is C ′ given by 11, 2, 12, 3, 13, 4, 14, 5, 16, 15, 6, 11, 7, 12, 8, 13, 9, 14, 16, 10, 15, 1, 11, 2, 12, 3, 13, 4, 14, 5, 16, 15, 6, 11, 7, 12, 8, 13, 9, 14, 16, 10, 15. Because φ(C) = 13 and φ(C ′ ) = 12 the two codes are not isomorphic.…”

Symmetry Implies Isomorphism for Certain Maximum Length Circuit Codes

“…where d I(d) (x, y) and d C (x, y) denote the minimum path length between vertices x and y in I(d) and C, respectively. Computing the maximum length of a (d, k) circuit code, K(d, k), for a given dimension d and spread k is an extremely challenging computational problem, and significant analysis is required to make the problem tractable even for small values of d and k [4,3,5]. Exact formulas for K(d, k) (for particular infinite families of (d, k) pairs) are exceedingly rare.…”