MinT - Best Known (89, 89+73, s)-Nets in Base 3

Best Known (89, 89+73, s)-Nets in Base 3

(89, 89+73, 80)-Net over F₃ — Constructive and digital

Digital (89, 162, 80)-net over F₃, using

trace code for nets [i] based on digital (8, 81, 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]

(89, 89+73, 112)-Net over F₃ — Digital

Digital (89, 162, 112)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+73, 936)-Net in Base 3 — Upper bound on s

There is no (89, 162, 937)-net in base 3, because

1 times m-reduction [i] would yield (89, 161, 937)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66379 888501 506013 361065 641528 065025 217185 923594 567863 674923 029772 001196 614545 > 3¹⁶¹ [i]