MinT - Best Known (166, 166+81, s)-Nets in Base 3

Best Known (166, 166+81, s)-Nets in Base 3

(166, 166+81, 168)-Net over F₃ — Constructive and digital

Digital (166, 247, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 51, 20)-net over F₃, using
  - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
2. digital (115, 196, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 98, 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]

(166, 166+81, 402)-Net over F₃ — Digital

Digital (166, 247, 402)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 166+81, 6737)-Net in Base 3 — Upper bound on s

There is no (166, 247, 6738)-net in base 3, because

1 times m-reduction [i] would yield (166, 246, 6738)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2355 538409 221893 927997 437474 821356 819695 493795 128283 898685 000531 172946 456249 370369 850680 120109 709287 316998 453839 689361 > 3²⁴⁶ [i]