MinT - Best Known (236−75, 236, s)-Nets in Base 3

Best Known (236−75, 236, s)-Nets in Base 3

(236−75, 236, 176)-Net over F₃ — Constructive and digital

Digital (161, 236, 176)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 52, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (109, 184, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 92, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(236−75, 236, 427)-Net over F₃ — Digital

Digital (161, 236, 427)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−75, 236, 7820)-Net in Base 3 — Upper bound on s

There is no (161, 236, 7821)-net in base 3, because

1 times m-reduction [i] would yield (161, 235, 7821)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13327 568681 934610 512658 422205 078977 024654 703081 498233 972614 122910 236696 487037 488500 911198 585044 202148 991860 348843 > 3²³⁵ [i]