MinT - Best Known (187−77, 187, s)-Nets in Base 3

Best Known (187−77, 187, s)-Nets in Base 3

(187−77, 187, 128)-Net over F₃ — Constructive and digital

Digital (110, 187, 128)-net over F₃, using

7 times m-reduction [i] based on digital (110, 194, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 97, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(187−77, 187, 162)-Net over F₃ — Digital

Digital (110, 187, 162)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−77, 187, 1588)-Net in Base 3 — Upper bound on s

There is no (110, 187, 1589)-net in base 3, because

1 times m-reduction [i] would yield (110, 186, 1589)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55546 416966 905772 561251 245898 734660 928512 292210 284821 788492 556566 530776 640107 303323 720865 > 3¹⁸⁶ [i]