MinT - Best Known (138−47, 138, s)-Nets in Base 3

Best Known (138−47, 138, s)-Nets in Base 3

(138−47, 138, 156)-Net over F₃ — Constructive and digital

Digital (91, 138, 156)-net over F₃, using

trace code for nets [i] based on digital (22, 69, 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]

(138−47, 138, 219)-Net over F₃ — Digital

Digital (91, 138, 219)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−47, 138, 3254)-Net in Base 3 — Upper bound on s

There is no (91, 138, 3255)-net in base 3, because

1 times m-reduction [i] would yield (91, 137, 3255)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 233604 616562 426629 946580 760672 071362 212392 462337 231827 651078 362931 > 3¹³⁷ [i]