MinT - Best Known (70, 70+37, s)-Nets in Base 3

Best Known (70, 70+37, s)-Nets in Base 3

(70, 70+37, 128)-Net over F₃ — Constructive and digital

Digital (70, 107, 128)-net over F₃, using

7 times m-reduction [i] based on digital (70, 114, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 57, 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]

(70, 70+37, 170)-Net over F₃ — Digital

Digital (70, 107, 170)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+37, 2419)-Net in Base 3 — Upper bound on s

There is no (70, 107, 2420)-net in base 3, because

1 times m-reduction [i] would yield (70, 106, 2420)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 377 601841 350294 838276 606773 615811 764579 215160 006105 > 3¹⁰⁶ [i]