MinT - Best Known (71, 71+63, s)-Nets in Base 3

Best Known (71, 71+63, s)-Nets in Base 3

(71, 71+63, 61)-Net over F₃ — Constructive and digital

Digital (71, 134, 61)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 44, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (27, 90, 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]

(71, 71+63, 85)-Net over F₃ — Digital

Digital (71, 134, 85)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+63, 661)-Net in Base 3 — Upper bound on s

There is no (71, 134, 662)-net in base 3, because

1 times m-reduction [i] would yield (71, 133, 662)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2879 102544 339653 617234 489316 031280 353494 879937 464176 256627 118825 > 3¹³³ [i]