The standard binary reflected Gray code produces a permutation of the sequence of integers (0, 1,…, n − 1), where n is a power of 2, such that the binary representation of each integer in the permuted sequence differs from the binary representation of the preceding integer in exactly one bit. In an earlier paper, we presented two methods to compute binary dense Gray codes, which extend the possible values of n to the set of all positive integers while preserving both the Gray-code property — only one bit changes between each pair of consecutive integers — and the denseness property — the sequence contains exactly the n integers 0 to n − 1. This paper generalizes our method for binary dense Gray codes to arbitrary radices that may be either a single fixed radix for all digits or mixed radices, so that each digit may have a different radix. That is, we show how to produce a permutation of (0, 1, …, n−1) represented in any set of radices, such that the representation of each number differs from the representation of the preceding number in exactly one digit, and the values of these digits differ by exactly 1. We provide a simple formula for this permuation, which we can use to quickly compute a Hamiltonian path for a dynamic array of n nodes, where the nodes are added and deleted in order along the k dimensions of a grid network.

/pdf/dense-gray-codes-in-mixed-radices-5bnrrs8euy.pdf

Dense gray codes in mixed radices

The Gray code is a sequence of n consecutive binary numbers arranged so that adjacent numbers in the sequence differ by 1 and that each number appears in the sequence exactly once. A Gray code is considered cyclic if the first and last numbers in the sequence differ in only one digit position, and those digits have a difference of exactly 1. In this thesis, numbers in a Gray-code sequence may be modular cyclic, meaning that in radix r, two consecutive numbers may vary in a digit with values 0 and r − 1. This thesis focuses on methods to generate mixed-radix Gray codes. Mixed-radix representation refers to any k-tuple of radices r = (rk−1, rk−2, . . . , r0). In a mixed-radix representation, each digit position can correspond to a different numerical base. In this thesis, we examine methods to generate two types of cyclic mixed-radix Gray codes: full and dense. Given pi = ∏i j=0 rj and p−1 = 1, a Gray code is considered full if n = pk−1, and a Gray code is considered dense if n is strictly less than pk−1. Any full or dense Gray code sequence is a permutation of the sequence 〈0, 1, . . . , n − 1〉, with each number appearing exactly once. We give a constant-time algorithm to generate each digit of each number in a cyclic mixed-radix full Gray code for any set of radices r. This thesis also provides a new case in which it is impossible to compute a cyclic mixed-radix dense Gray code. For all r and n for which we do not have proofs that a cyclic Gray code cannot exist and where rk−1 ≥ 3 or n is even, we show a linear-time method to find a cyclic mixed-radix dense Gray code. When rk−1 = 2 and n is odd, we provide a conjecture regarding the possibility of a cyclic mixed-radix dense Gray code for a mixed-radix tuple r.

/pdf/full-and-dense-cyclic-gray-codes-in-mixed-radices-97phc851m9.pdf

Full and Dense Cyclic Gray codes in Mixed Radices

The standard binary reflected Gray code gives a sequence of binary numbers in the range 0 to n − 1, where n is a power of 2, such that each number in the sequence differs from the preceding number in only one bit. We present two methods to compute Gray codes containing exactly n numbers in the range 0 to n − 1—that is, a permutation of 〈0, 1, …, n − 1〉 in which each number differs from the preceding number in only one bit—where n is unconstrained. The first method produces a Gray code that is not cyclic: the first and last numbers in the sequence differ in more than one bit. The second method produces a cyclic Gray code if n is even, so that the first and last numbers differ in only one bit, at the expense of a slightly more complicated procedure. Both methods are based on the standard binary reflected Gray code and, as in the binary reflected Gray code, each number in the output sequence can be computed in a constant number of word operations given just its index in the sequence.

Jessica C. Fan

Papers

Dense Gray codes, or easy ways to generate cyclic and non-cyclic Gray codes for the first n whole numbers

Dense gray codes in mixed radices