MinT - Best Known (133, 133+113, s)-Nets in Base 3

Best Known (133, 133+113, s)-Nets in Base 3

(133, 133+113, 86)-Net over F₃ — Constructive and digital

Digital (133, 246, 86)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (32, 88, 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]
2. digital (45, 158, 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]

(133, 133+113, 152)-Net over F₃ — Digital

Digital (133, 246, 152)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+113, 1272)-Net in Base 3 — Upper bound on s

There is no (133, 246, 1273)-net in base 3, because

1 times m-reduction [i] would yield (133, 245, 1273)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 788 384971 967088 049919 970227 081315 643586 131491 280593 915745 227582 751503 995325 070951 877557 728605 313787 463571 742142 708833 > 3²⁴⁵ [i]