MinT - Best Known (220−107, 220, s)-Nets in Base 3

Best Known (220−107, 220, s)-Nets in Base 3

(220−107, 220, 76)-Net over F₃ — Constructive and digital

Digital (113, 220, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 68, 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 (45, 152, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(220−107, 220, 120)-Net over F₃ — Digital

Digital (113, 220, 120)-net over F₃, using

net from sequence [i] based on digital (113, 119)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(220−107, 220, 913)-Net in Base 3 — Upper bound on s

There is no (113, 220, 914)-net in base 3, because

1 times m-reduction [i] would yield (113, 219, 914)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 324 949984 974055 585829 547653 802435 725876 415963 908530 603332 699620 176491 503082 809014 399622 564670 060607 823325 > 3²¹⁹ [i]