Weighted-covering code

Description

A \(q\)-ary code for which balls of some radius centered at its codewords provide a weighted covering of the Hamming space.

Let the outer or weight distribution of a \(q\)-ary string \(x\) with respect to a \(q\)-ary code \(C\) be \(A(x) = \left( A_0(x),A_1(x),\cdots,A_n(x) \right)\), where \begin{align} A_j(x) = \left|\{ c \in C~\text{such that}~ D(c,x)=j \}\right|~, \tag*{(1)}\end{align} and \(D\) is the Hamming distance. Given a tuple \(m=(m_1,m_2,\cdots,m_n)\) of rational numbers, the \(m\)-density of the code at \(x\) is \begin{align} \theta(x) = \sum_{j=0}^n m_j A_j(x)~. \tag*{(2)}\end{align}

A code is an \(m\)-weighted covering if \(\theta(x)\geq1\) for all strings \(x\in GF(q)^n\). The \(m\)-covering radius \(r\) is the largest \(j\) for which \(m_j\) is nonzero.