MinT - Best Known (179−61, 179, s)-Nets in Base 3

Best Known (179−61, 179, s)-Nets in Base 3

(179−61, 179, 156)-Net over F₃ — Constructive and digital

Digital (118, 179, 156)-net over F₃, using

13 times m-reduction [i] based on digital (118, 192, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 96, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(179−61, 179, 271)-Net over F₃ — Digital

Digital (118, 179, 271)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(179−61, 179, 4050)-Net in Base 3 — Upper bound on s

There is no (118, 179, 4051)-net in base 3, because

1 times m-reduction [i] would yield (118, 178, 4051)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 479257 282990 403141 171542 939085 198338 751518 174404 429043 977695 399693 321215 795861 507021 > 3¹⁷⁸ [i]