MinT - Best Known (228−74, 228, s)-Nets in Base 3

Best Known (228−74, 228, s)-Nets in Base 3

(228−74, 228, 167)-Net over F₃ — Constructive and digital

Digital (154, 228, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 46, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (108, 182, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 91, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(228−74, 228, 387)-Net over F₃ — Digital

Digital (154, 228, 387)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−74, 228, 6345)-Net in Base 3 — Upper bound on s

There is no (154, 228, 6346)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 078754 850783 243674 109059 869260 229570 198147 124470 864984 328037 250014 457010 215746 428989 695919 871179 076997 448621 > 3²²⁸ [i]