MathDB

Let n \equal{} 2k \minus{} 1 where

k \geq 6

is an integer. Let

T

be the set of all n\minus{}tuples

(x_1, x_2, \ldots, x_n)

where

x_i \in \{0,1\}

\forall i \equal{} \{1,2, \ldots, n\} For x \equal{} (x_1, x_2, \ldots, x_n) \in T and y \equal{} (y_1, y_2, \ldots, y_n) \in T let

d(x,y)

denote the number of integers

j

with

1 \leq j \leq n

such that

x_i \neq x_j

, in particular d(x,x) \equal{} 0. Suppose that there exists a subset

S

T

with

2^k

elements that has the following property: Given any element

x \in T,

there is a unique element

y \in S

with

d(x, y) \leq 3.

Prove that n \equal{} 23.

Problem Statement