MinT - Best Known (127−59, 127, s)-Nets in Base 3

Best Known (127−59, 127, s)-Nets in Base 3

(127−59, 127, 60)-Net over F₃ — Constructive and digital

Digital (68, 127, 60)-net over F₃, using

1 times m-reduction [i] based on digital (68, 128, 60)-net over F₃, using
- trace code for nets [i] based on digital (4, 64, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]

(127−59, 127, 84)-Net over F₃ — Digital

Digital (68, 127, 84)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−59, 127, 662)-Net in Base 3 — Upper bound on s

There is no (68, 127, 663)-net in base 3, because

1 times m-reduction [i] would yield (68, 126, 663)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 344479 490102 404798 889617 157706 518328 757921 877691 858060 906935 > 3¹²⁶ [i]