MinT - Best Known (72, 72+49, s)-Nets in Base 3

Best Known (72, 72+49, s)-Nets in Base 3

(72, 72+49, 80)-Net over F₃ — Constructive and digital

Digital (72, 121, 80)-net over F₃, using

7 times m-reduction [i] based on digital (72, 128, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 64, 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]

(72, 72+49, 119)-Net over F₃ — Digital

Digital (72, 121, 119)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+49, 1167)-Net in Base 3 — Upper bound on s

There is no (72, 121, 1168)-net in base 3, because

1 times m-reduction [i] would yield (72, 120, 1168)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1805 483975 296421 622810 087150 494929 320528 655597 413969 038337 > 3¹²⁰ [i]