MinT - Best Known (113−56, 113, s)-Nets in Base 3

Best Known (113−56, 113, s)-Nets in Base 3

(113−56, 113, 52)-Net over F₃ — Constructive and digital

Digital (57, 113, 52)-net over F₃, using

2 times m-reduction [i] based on digital (57, 115, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 42, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (15, 73, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]

(113−56, 113, 64)-Net over F₃ — Digital

Digital (57, 113, 64)-net over F₃, using

t-expansion [i] based on digital (49, 113, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(113−56, 113, 449)-Net in Base 3 — Upper bound on s

There is no (57, 113, 450)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 870124 876026 760354 231187 065422 783172 944656 926071 933977 > 3¹¹³ [i]