The Algebraic Degree of Perfect Binary Codes

In this communication, we give a complete characterization for the range of the algebraic degree of perfect binary codes. For 1 -perfect binary codes, we show that for any
d
in the algebraic degree range, there are Vasil'ev codes with algebraic degree
d
. It is also shown that the algebraic degree of 1-perfect binary codes is independent of the rank and the dimension of the kernel.