MinT - Best Known (187−63, 187, s)-Nets in Base 3

Best Known (187−63, 187, s)-Nets in Base 3

(187−63, 187, 156)-Net over F₃ — Constructive and digital

Digital (124, 187, 156)-net over F₃, using

17 times m-reduction [i] based on digital (124, 204, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 102, 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]

(187−63, 187, 291)-Net over F₃ — Digital

Digital (124, 187, 291)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−63, 187, 4495)-Net in Base 3 — Upper bound on s

There is no (124, 187, 4496)-net in base 3, because

1 times m-reduction [i] would yield (124, 186, 4496)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55627 800247 812579 164236 384709 203841 829064 565746 025197 596372 016363 336666 650051 187094 540993 > 3¹⁸⁶ [i]