MinT - Best Known (178−79, 178, s)-Nets in Base 3

Best Known (178−79, 178, s)-Nets in Base 3

(178−79, 178, 80)-Net over F₃ — Constructive and digital

Digital (99, 178, 80)-net over F₃, using

4 times m-reduction [i] based on digital (99, 182, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 91, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(178−79, 178, 126)-Net over F₃ — Digital

Digital (99, 178, 126)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−79, 178, 1088)-Net in Base 3 — Upper bound on s

There is no (99, 178, 1089)-net in base 3, because

1 times m-reduction [i] would yield (99, 177, 1089)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 851234 511663 306957 793009 153348 644268 135310 492614 267497 066920 268397 269601 958480 546059 > 3¹⁷⁷ [i]