MinT - Best Known (138, 138+77, s)-Nets in Base 3

Best Known (138, 138+77, s)-Nets in Base 3

(138, 138+77, 156)-Net over F₃ — Constructive and digital

Digital (138, 215, 156)-net over F₃, using

17 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+77, 273)-Net over F₃ — Digital

Digital (138, 215, 273)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+77, 3616)-Net in Base 3 — Upper bound on s

There is no (138, 215, 3617)-net in base 3, because

1 times m-reduction [i] would yield (138, 214, 3617)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 280443 598254 336517 733315 569940 806257 921162 373502 274752 939171 696434 273394 553829 318944 969793 012588 788377 > 3²¹⁴ [i]