On Binary Linear (16, 8) Code with the Maximum Minimum Distance

Abstract

It is known that the maximum value of minimum distances of binary linear (16, 8) codes is 5. This paper shows that the binary linear (16, 8) code with the minimum distance 5 is unique up to the permutation of bit positions. It is also confirmed that there is no binary linear (16, 8) code with the minimum distance 6, which implies there is no binary linear (15, 8) code with the minimum distance 5, although there is a well-known binary non-linear (15, 8) code of Preparata with the minimum distance 5. The above facts are obtained by finding novel search algorithms in parity-check matrices for binary linear codes.