Distribution of the weights of the dual of the Melas code
The title of this paper suggests that the weight distribution of the dual of the Melas code is determined. However the author only gives a property of this weight distribution, using some deep results of Deligne and Katz. More precisely if \(f\) is some testfunction set \[ g_ q(w)=f((q-1- 2w)/2\sqrt{q})\text{ for} w \in [(q-1-2\sqrt{q})/2,(q-1+2\sqrt{q})/2] \] and \(\phi_ q(w)=(1/\pi q)\sqrt{4q-(q-1-2w)^ 2}\). Then: \[ (1/q^ 2)\sum_{x\in C_{k\ell}(q)}g_ q(w(x))=\int^{w^+}_{w_-}g_ q(w)\phi_ q(w)dw+O(1/\sqrt{q})\text{ when} q\to \infty \] and the constant of the O-term only depends on f. The paper contains the following misprints: p.104: middle of the page: \(z(x)=(2w(x)-(q-1))/2\sqrt{q}\) should read \(z(x)=((q-1)- 2w(x))/2\sqrt{q};\) p. 105: first formula: \(q-1+w_{k\ell}(a)\) should read: \(q-1-w_{k\ell}(a);\) p. 105: formula following Remark should read: \(w=(q-1-2\sqrt{q}z)/2;\) p. 106: first formula should read: \(\phi_ q(w)=(1/\pi q)\sqrt{4q-(q-1-2w)^ 2},\) second formula should read: \(g_ q(w)=f((q-1-2w)/2Vq).\)
scientific article; zbMATH DE number 4145827