Questions tagged [gray-codes]

A Gray code is a cyclic ordering of binary strings so that two successive values differ in only one bit (a Hamming distance of 1).

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Multiset of Hamming distances for a tour of all subsets

Consider a list of all the subsets of $\{1,\ldots,n\}$, in any order. Using binary notation, one such list for $n=3$ is $$ 010, 100, 110, 011, 000, 111, 001, 101. $$ Now consider the Hamming distance ...
Brendan McKay's user avatar
2 votes
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Counting matchings in middle levels of the Boolean lattice

Let $k$ be a nonnegative integer and consider $C_k$, the set of all subsets $A$ of size $k$ in $[2k+1]=\{1,2,\ldots,2k+1\}$ as well as $C_{k+1}$, the set of all subsets $B$ of size $k+1$ in $[2k+1]$. ...
Abdelmalek Abdesselam's user avatar
7 votes
3 answers

"Gray code" for building teams

Motivation. In a team of $n$ people, we had the task to build subteams of a fixed size $k<n$ such that every day, $1$ person of the subteam is replaced by another person in the team, but not in the ...
Dominic van der Zypen's user avatar
2 votes
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Balanced Gray codes for powers of 2

All of the binary 4-bit cyclic balanced Gray code sequences can be formed from simple reversals, bit-permutations, and circular shifts of the one Wikipedia example: ...
bobuhito's user avatar
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12 votes
3 answers

Maximal distance of $2d+1$ points on a sphere

If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
M. Winter's user avatar
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6 votes
1 answer

Are Gray codes in $\{0,1\}^n$ isomorphic?

Let $n\in\mathbb{N}$ be a positive integer. Two elements of $\{0,1\}^n$ form an edge if and only if their Hamming distance equals $1$. It is known that $\{0,1\}^n$ endowed with this graph structure ...
Dominic van der Zypen's user avatar
2 votes
1 answer

New Perfect 2-bit Error Correction Code - Are there any other?

I have recently looked through perfect error correcting codes and found the Hamming(7,3) and Golay(23,7). Using a computer program I have found a new 2 bit perfect error correcting code: Code(90, 2). ...
RobertB.'s user avatar
1 vote
1 answer

Hamiltonian Path through $n$-bit strings with maximum number of $0\mapsto 1$ transitions

Let $G_n$ be the complete graph whose vertices are the $2^n$ $n$-bit strings. Let $H_n$ denote the Hamiltonian path through $G_n$ that uses the maximum number of edges that correspond to a single bit ...
Stephan Mertens's user avatar
1 vote
1 answer

How many non-overlapping k-hop neighborhoods can be uniquely colored on an $N$-dimensional hypercube?

Imagine I have a $N$-dimensional hypercube. My aim is to distinctly color as many non-overlapping $k$-hop neighborhoods as possible (i.e. sets of vertices connected by a Manhattan distance of at most ...
StoicPorpoise's user avatar
3 votes
1 answer

Sprague-Grundy sequence for the ruler game

Consider the game "Ruler", which is defined as follows. We start with finitely many coins in a line. A move in this game consists of turning over any number of coins, but they must be consecutive, ...
Michael Lugo's user avatar
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