MinT - Best Known (124, 124+65, s)-Nets in Base 3

Best Known (124, 124+65, s)-Nets in Base 3

(124, 124+65, 156)-Net over F₃ — Constructive and digital

Digital (124, 189, 156)-net over F₃, using

15 times m-reduction [i] based on digital (124, 204, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 102, 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]

(124, 124+65, 276)-Net over F₃ — Digital

Digital (124, 189, 276)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+65, 4032)-Net in Base 3 — Upper bound on s

There is no (124, 189, 4033)-net in base 3, because

1 times m-reduction [i] would yield (124, 188, 4033)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 501959 782858 693253 686080 538288 360114 516737 871941 967985 969209 149303 670088 286394 768623 361665 > 3¹⁸⁸ [i]