MinT - Best Known (242−113, 242, s)-Nets in Base 3

Best Known (242−113, 242, s)-Nets in Base 3

(242−113, 242, 85)-Net over F₃ — Constructive and digital

Digital (129, 242, 85)-net over F₃, using

1 times m-reduction [i] based on digital (129, 243, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 84, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
  2. digital (45, 159, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(242−113, 242, 143)-Net over F₃ — Digital

Digital (129, 242, 143)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−113, 242, 1172)-Net in Base 3 — Upper bound on s

There is no (129, 242, 1173)-net in base 3, because

1 times m-reduction [i] would yield (129, 241, 1173)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 785879 336849 237346 450887 312172 701416 784094 130578 719961 107269 169574 302903 938239 503295 032500 550663 833562 149970 100897 > 3²⁴¹ [i]