Binary code

A binary code \(C\) corrects \(t\) errors in the Hamming distance if \begin{align} \forall x \in C~,~D(x,x+y) < D(x' , x+y) \tag*{(1)}\end{align} for all codewords \(x' \neq x\) and all \(y\) such that \(|y|=t\), where \(D\) is the Hamming distance and \(|y| = D(y,0) \). A code corrects \(t\) errors if and only if \(d \geq 2t+1\), i.e., a code corrects errors on \(t \leq \left\lfloor (d-1)/2 \right\rfloor\) coordinates. The number of correctable errors is called the decoding radius, and it is upper bounded by half of the distance. In addition, a code detects errors on up to \(d-1\) coordinates, and corrects erasure errors on up to \(d-1\) coordinates.

Performance of binary codes can also be measured with respect to deletions and insertions of bits into the codewords. In this case, the metric measuring distance of an error word to the nearest codeword is the deletion distance: given vectors \(u,v\), this distance is one-half the smallest number of deletions and insertions needed to change \(u\) to \(v\). A code \(C\) corrects \(e\) delections if all codewords are separated by at least \(e+1\) in the deletion distance [1].