MinT - Best Known (178−84, 178, s)-Nets in Base 3

Best Known (178−84, 178, s)-Nets in Base 3

(178−84, 178, 72)-Net over F₃ — Constructive and digital

Digital (94, 178, 72)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 68, 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 (26, 110, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃ (see above)

(178−84, 178, 107)-Net over F₃ — Digital

Digital (94, 178, 107)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−84, 178, 827)-Net in Base 3 — Upper bound on s

There is no (94, 178, 828)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 471993 542567 655228 773716 911119 399363 793005 536426 574187 003253 751453 458021 152252 072841 > 3¹⁷⁸ [i]