MinT - Best Known (157−67, 157, s)-Nets in Base 3

Best Known (157−67, 157, s)-Nets in Base 3

(157−67, 157, 80)-Net over F₃ — Constructive and digital

Digital (90, 157, 80)-net over F₃, using

7 times m-reduction [i] based on digital (90, 164, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 82, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(157−67, 157, 127)-Net over F₃ — Digital

Digital (90, 157, 127)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−67, 157, 1153)-Net in Base 3 — Upper bound on s

There is no (90, 157, 1154)-net in base 3, because

1 times m-reduction [i] would yield (90, 156, 1154)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 276 511448 412880 015931 844657 404342 102508 611088 818330 794648 587697 961722 601925 > 3¹⁵⁶ [i]