MinT - Best Known (222−109, 222, s)-Nets in Base 3

Best Known (222−109, 222, s)-Nets in Base 3

(222−109, 222, 75)-Net over F₃ — Constructive and digital

Digital (113, 222, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 81, 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]
2. digital (32, 141, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]

(222−109, 222, 120)-Net over F₃ — Digital

Digital (113, 222, 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]

(222−109, 222, 887)-Net in Base 3 — Upper bound on s

There is no (113, 222, 888)-net in base 3, because

1 times m-reduction [i] would yield (113, 221, 888)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2804 931155 663546 284710 415252 595561 036173 383611 997475 936368 792323 611959 141660 843067 935930 351402 798413 449681 > 3²²¹ [i]