MinT - Best Known (138, 138+82, s)-Nets in Base 3

Best Known (138, 138+82, s)-Nets in Base 3

(138, 138+82, 156)-Net over F₃ — Constructive and digital

Digital (138, 220, 156)-net over F₃, using

12 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, 138+82, 249)-Net over F₃ — Digital

Digital (138, 220, 249)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+82, 2890)-Net in Base 3 — Upper bound on s

There is no (138, 220, 2891)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 926 997163 129762 459719 297534 137832 811304 003182 389688 646128 101350 946346 000652 735950 071260 374122 797246 617095 > 3²²⁰ [i]