MinT - Best Known (129, 129+59, s)-Nets in Base 3

Best Known (129, 129+59, s)-Nets in Base 3

(129, 129+59, 164)-Net over F₃ — Constructive and digital

Digital (129, 188, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (93, 152, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 76, 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]

(129, 129+59, 365)-Net over F₃ — Digital

Digital (129, 188, 365)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+59, 6933)-Net in Base 3 — Upper bound on s

There is no (129, 188, 6934)-net in base 3, because

1 times m-reduction [i] would yield (129, 187, 6934)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 167276 175095 412996 221981 021042 156303 350597 923840 799579 877232 130454 866894 606053 589592 139861 > 3¹⁸⁷ [i]