MinT - Best Known (126, 126+63, s)-Nets in Base 3

Best Known (126, 126+63, s)-Nets in Base 3

(126, 126+63, 156)-Net over F₃ — Constructive and digital

Digital (126, 189, 156)-net over F₃, using

19 times m-reduction [i] based on digital (126, 208, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 104, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(126, 126+63, 303)-Net over F₃ — Digital

Digital (126, 189, 303)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+63, 4828)-Net in Base 3 — Upper bound on s

There is no (126, 189, 4829)-net in base 3, because

1 times m-reduction [i] would yield (126, 188, 4829)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 502373 156532 033668 310479 598917 201535 867860 141777 099506 704979 522366 212053 148498 613115 841563 > 3¹⁸⁸ [i]