MinT - Best Known (72, 72+45, s)-Nets in Base 3

Best Known (72, 72+45, s)-Nets in Base 3

(72, 72+45, 128)-Net over F₃ — Constructive and digital

Digital (72, 117, 128)-net over F₃, using

1 times m-reduction [i] based on digital (72, 118, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 59, 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]

(72, 72+45, 133)-Net over F₃ — Digital

Digital (72, 117, 133)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+45, 1463)-Net in Base 3 — Upper bound on s

There is no (72, 117, 1464)-net in base 3, because

1 times m-reduction [i] would yield (72, 116, 1464)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 504067 322959 796932 822853 593930 385899 951222 140189 838673 > 3¹¹⁶ [i]