Polynomial description of binary linear codes and related properties (Q1181004)

In this important and elegant paper the author introduces a polynomial which by means of a trace function is related to a binary projective linear code (i.e. a binary linear code without repeated columns). Properties on parameters, weights, automorphism group of the codes can be translated directly into properties of the polynomial and vice versa. A fundamental point in studying the properties of the codes and related polynomial is the number of solutions to the equation \(\text{tr}(f(x)+ax)=1\), where \(f\) is fixed, \(a\) runs through \(GF(2^ k)\) and tr is the trace function from \(GF(2^ k)\) to \(GF(2)\). Some minor details: On page 122 Examples 2 line 7 in the formula for \(F(x)\) and \(h(x)\), the constant term \(a\) is missing. On page 123 line 4: \(T_ 1\) should read \(T^*\). In the proof of Theorem 3.6 a, I do not see why this case is a direct consequence of Lemma 3.2 however by applying Theorem 3.5 and the results of Example 3.2.1 I arrived at the desired result. In Example 12 in the sequence 5,3,5,5,1,1 there is a mistake, it should be 5,3,5,1,5,3. Therefore \(\pi(f)=6\) (and not 8 as stated).