A Riemann hypothesis analogue for self-dual codes (Q2717199)

For an arbitrary \(q\)-ary linear code, a polynomial that is uniquely determined by the weight enumerator of the code is defined. It is called the zeta polynomial and contains precisely the same information as the weight enumerator, but presented in a form that resembles the classical zeta function. A self-dual code is referred to as a code that satisfies the analogue of the Riemann hypothesis (has the RH property) if all the reciprocal zeros of its zeta polynomial have the same absolute value (\(=\sqrt{q}\)). NEWLINENEWLINENEWLINEIt is proved in the paper that any \([12,6,4]\) self-dual code is such a code. Also, an upper bound for minimum distances of a class of self-dual codes is given. NEWLINENEWLINENEWLINESince the \([8,4,4]\) extended Hamming code does not have the RH property, the author poses the following open problem: Prove or disprove that all extremal weight enumerators have the RH property. NEWLINENEWLINENEWLINETwo other open problems are given at the end of the paper.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00079].