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We present the domain-wall fermion operator which is reflection symmetric in the fifth dimension, with the approximate sign function $ S(H) $ of the effective 4-dimensional Dirac operator satisfying the bound $ |1-S(\lambda)| \le 2 d_Z $ for $ \lambda^2 \in [\lambda_{min}^2, \lambda_{max}^2] $, where $ d_Z $ is the maximum deviation $ | 1- \sqrt{x} R_Z(x) |_{\rm max} $ of the Zolotarev optimal rational polynomial $ R_Z(x) $ of $ 1/\sqrt{x} $ for $ x \in [1, \lambda_{max}^2/\lambda_{min}^2] $, with degrees $ (n-1,n) $ for $ N_s = 2n $, and $ (n,n) $ for $ N_s = 2 n + 1 $.
E. I. Zolotarev, Zap. Imp. Akad. Nauk. St. Petersburg, 30 (1877), no. 5; reprinted in his Collected works, Vol. 2, Izdat, Akad. Nauk SSSR, Moscow, 1932, p. 1-59
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