MinT - Best Known (178−59, 178, s)-Nets in Base 3

Best Known (178−59, 178, s)-Nets in Base 3

(178−59, 178, 156)-Net over F₃ — Constructive and digital

Digital (119, 178, 156)-net over F₃, using

16 times m-reduction [i] based on digital (119, 194, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 97, 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]

(178−59, 178, 294)-Net over F₃ — Digital

Digital (119, 178, 294)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−59, 178, 4737)-Net in Base 3 — Upper bound on s

There is no (119, 178, 4738)-net in base 3, because

1 times m-reduction [i] would yield (119, 177, 4738)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 824436 437569 240350 889042 294655 107320 909279 348441 706123 554657 187405 550653 487749 988269 > 3¹⁷⁷ [i]