MinT - Best Known (125−36, 125, s)-Nets in Base 9

Best Known (125−36, 125, s)-Nets in Base 9

(125−36, 125, 768)-Net over F₉ — Constructive and digital

Digital (89, 125, 768)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (68, 104, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 52, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(125−36, 125, 4466)-Net over F₉ — Digital

Digital (89, 125, 4466)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−36, 125, 3996897)-Net in Base 9 — Upper bound on s

There is no (89, 125, 3996898)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 190684 304099 695209 967858 232644 536736 365499 074148 072616 261656 144216 913671 041132 210780 057130 481131 800613 816389 860772 814305 > 9¹²⁵ [i]