MinT - Best Known (127−38, 127, s)-Nets in Base 9

Best Known (127−38, 127, s)-Nets in Base 9

(127−38, 127, 750)-Net over F₉ — Constructive and digital

Digital (89, 127, 750)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (70, 108, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 54, 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]

(127−38, 127, 3471)-Net over F₉ — Digital

Digital (89, 127, 3471)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−38, 127, 2368542)-Net in Base 9 — Upper bound on s

There is no (89, 127, 2368543)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 445392 822139 145159 894298 137029 394177 638897 052272 708772 065306 938158 101642 957938 404218 806578 036032 690395 954031 144178 837097 > 9¹²⁷ [i]