MinT - Best Known (138, 231, s)-Nets in Base 3

Best Known (138, 231, s)-Nets in Base 3

(138, 231, 156)-Net over F₃ — Constructive and digital

Digital (138, 231, 156)-net over F₃, using

1 times m-reduction [i] based on digital (138, 232, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 116, 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]

(138, 231, 208)-Net over F₃ — Digital

Digital (138, 231, 208)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 231, 2141)-Net in Base 3 — Upper bound on s

There is no (138, 231, 2142)-net in base 3, because

1 times m-reduction [i] would yield (138, 230, 2142)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55 062243 291916 783929 552836 724142 874680 797586 224525 662894 922883 867225 309336 890462 749537 757890 186511 396661 109685 > 3²³⁰ [i]