MinT - Best Known (228−103, 228, s)-Nets in Base 3

Best Known (228−103, 228, s)-Nets in Base 3

(228−103, 228, 85)-Net over F₃ — Constructive and digital

Digital (125, 228, 85)-net over F₃, using

3 times m-reduction [i] based on digital (125, 231, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 80, 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 (45, 151, 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]

(228−103, 228, 149)-Net over F₃ — Digital

Digital (125, 228, 149)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−103, 228, 1270)-Net in Base 3 — Upper bound on s

There is no (125, 228, 1271)-net in base 3, because

1 times m-reduction [i] would yield (125, 227, 1271)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 104192 322714 891408 292739 821255 092008 929079 586804 901632 182942 643405 603532 698839 452871 268716 279961 587826 690427 > 3²²⁷ [i]