MathDB

Given two integers

n\geq 1

and

q\geq 2

, let

A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}

. If

a=(a_1,\ldots ,a_n)

and

b=(b_1,\ldots ,b_n)

are two elements of

A

, let

\delta(a,b)=\#\{i:a_i\neq b_i\}

. Let further

t

be a non-negative integer and

B

a non-empty subset of

A

such that

\delta(a,b)\geq 2t+1

, whenever

a

and

b

are distinct elements of

B

. Prove that the two statements below are equivalent: a) For any

a\in A

, there is a unique

b\in B

, such that

\delta (a,b)\leq t

; b)

\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n

Problem Statement