MinT - Best Known (138, 138+79, s)-Nets in Base 3

Best Known (138, 138+79, s)-Nets in Base 3

(138, 138+79, 156)-Net over F₃ — Constructive and digital

Digital (138, 217, 156)-net over F₃, using

15 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+79, 263)-Net over F₃ — Digital

Digital (138, 217, 263)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+79, 3341)-Net in Base 3 — Upper bound on s

There is no (138, 217, 3342)-net in base 3, because

1 times m-reduction [i] would yield (138, 216, 3342)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 479678 664578 975795 256545 453875 218939 787130 504883 389544 128822 513507 908213 284867 263363 150544 825902 806201 > 3²¹⁶ [i]