MinT - Best Known (228−109, 228, s)-Nets in Base 3

Best Known (228−109, 228, s)-Nets in Base 3

(228−109, 228, 78)-Net over F₃ — Constructive and digital

Digital (119, 228, 78)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 80, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (39, 148, 42)-net over F₃, using
  - net from sequence [i] based on digital (39, 41)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(228−109, 228, 127)-Net over F₃ — Digital

Digital (119, 228, 127)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−109, 228, 1009)-Net in Base 3 — Upper bound on s

There is no (119, 228, 1010)-net in base 3, because

1 times m-reduction [i] would yield (119, 227, 1010)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 051203 620021 875126 282457 195623 371364 514840 870033 802460 298151 979998 077598 023207 129082 420322 536745 350856 548029 > 3²²⁷ [i]