MinT - Best Known (187−67, 187, s)-Nets in Base 3

Best Known (187−67, 187, s)-Nets in Base 3

(187−67, 187, 156)-Net over F₃ — Constructive and digital

Digital (120, 187, 156)-net over F₃, using

9 times m-reduction [i] based on digital (120, 196, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 98, 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]

(187−67, 187, 242)-Net over F₃ — Digital

Digital (120, 187, 242)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−67, 187, 3185)-Net in Base 3 — Upper bound on s

There is no (120, 187, 3186)-net in base 3, because

1 times m-reduction [i] would yield (120, 186, 3186)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55880 844146 017200 859169 791021 682762 258175 377045 008102 323946 480961 373368 680931 354043 134373 > 3¹⁸⁶ [i]