A binary perfect code of length 15 and codimension 0 (Q1329112)

Let \(C\) be a perfect binary nonlinear single error correcting code of length 15. The kernel of \(C\) is defined as \(\ker(C)= \{v\in F_ 2^{15}\mid v+c\in C\) for every element \(c\in C\}\) where \(F_ 2\) is the field of two elements. Let \(\langle C\rangle^ \perp\) denote the orthogonal code, the set of all elements of \(F_ 2^{15}\) orthogonal to all elements of \(C\). The codimension of \(C\), \(\text{cdim}(C)\), is the dimension of \(\langle C\rangle^ \perp\). Three new perfect single error correcting codes, \(C_ i\), are constructed with \(\dim (\ker (C_ i)) =i\), \(i=1,2,3\). It follows that the codimension of \(C_ 1\) is zero. The construction uses MDS codes of length 4, dimension 3 over \(F_ 4\).