On the structure of the binary (18,9,6) linear code

Let E_m denote the binary (m, m-1, 2) even weight code and ⊖ the Kronecker product operation. A linear code of length m×t containing E_m⊖(t,1,t) can be decoded efficiently using a Tanner graph and a trellis diagram together with the Wagner decoding algorithm. Thus, it is important to determine whether a given code contains such a subcode. We show that the (18,9,6) binary quadratic residue code does not contain E₆⊖(3,1,3). The maximum minimum distance of such a code is 5.