The Tutte Polynomial and Linear Codes

A linear code of length n and dimension k is called a [n, k]-code. If G is a r × n matrix over GF (q), whose rows form a basis for C, then G is a generator matrix for C. As with every matrix, there is an associated matroid, which in this case depends only on C. Hence the matroid is M(C). The dual code, C∗, of C, is defined by C∗ = {v ∈ GF (q)n|v · w = 0 ∀w ∈ C}. If C is a [n, k]-code, then C∗ is a [n, n− k]-code. Furthermore, M(C∗) ∼= M∗(C) [White (1986)] The members of a linear code are called codewords. If v is the codeword (v1, . . . , vn), then the weight of v is w(v) = |{i|vi 6= 0}|. The distance of C, denoted d(C), is inf v∈C−0 {w(v)}. It should be obvious that d is the size of a smallest bond in M(C), or the size of a smallest circuit in M∗(C).