MinT - Best Known (129−81, 129, s)-Nets in Base 4

Best Known (129−81, 129, s)-Nets in Base 4

(129−81, 129, 56)-Net over F₄ — Constructive and digital

Digital (48, 129, 56)-net over F₄, using

t-expansion [i] based on digital (33, 129, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(129−81, 129, 81)-Net over F₄ — Digital

Digital (48, 129, 81)-net over F₄, using

t-expansion [i] based on digital (46, 129, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(129−81, 129, 411)-Net in Base 4 — Upper bound on s

There is no (48, 129, 412)-net in base 4, because

1 times m-reduction [i] would yield (48, 128, 412)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 116429 247194 526149 177512 483346 182713 622092 532532 202827 827297 140697 922069 671300 > 4¹²⁸ [i]