MinT - Best Known (225−111, 225, s)-Nets in Base 3

Best Known (225−111, 225, s)-Nets in Base 3

(225−111, 225, 75)-Net over F₃ — Constructive and digital

Digital (114, 225, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 82, 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, 143, 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]

(225−111, 225, 120)-Net over F₃ — Digital

Digital (114, 225, 120)-net over F₃, using

t-expansion [i] based on digital (113, 225, 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]

(225−111, 225, 882)-Net in Base 3 — Upper bound on s

There is no (114, 225, 883)-net in base 3, because

1 times m-reduction [i] would yield (114, 224, 883)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75632 705992 679882 875452 803451 474432 856693 263754 292233 589582 561560 801963 119248 464389 290643 753477 524457 590115 > 3²²⁴ [i]