MinT - Best Known (81, 81+53, s)-Nets in Base 3

Best Known (81, 81+53, s)-Nets in Base 3

(81, 81+53, 128)-Net over F₃ — Constructive and digital

Digital (81, 134, 128)-net over F₃, using

2 times m-reduction [i] based on digital (81, 136, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 68, 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]

(81, 81+53, 138)-Net over F₃ — Digital

Digital (81, 134, 138)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+53, 1429)-Net in Base 3 — Upper bound on s

There is no (81, 134, 1430)-net in base 3, because

1 times m-reduction [i] would yield (81, 133, 1430)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2871 064671 214704 044748 683447 557903 281240 185990 893536 524707 107837 > 3¹³³ [i]