MinT - Best Known (88, 88+87, s)-Nets in Base 3

Best Known (88, 88+87, s)-Nets in Base 3

(88, 88+87, 65)-Net over F₃ — Constructive and digital

Digital (88, 175, 65)-net over F₃, using

5 times m-reduction [i] based on digital (88, 180, 65)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 61, 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]
  2. digital (27, 119, 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]

(88, 88+87, 91)-Net over F₃ — Digital

Digital (88, 175, 91)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(88, 88+87, 678)-Net in Base 3 — Upper bound on s

There is no (88, 175, 679)-net in base 3, because

1 times m-reduction [i] would yield (88, 174, 679)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 109916 728892 759941 976031 644770 749785 249874 896535 060584 541689 483492 660283 996653 735115 > 3¹⁷⁴ [i]