MinT - Best Known (164−87, 164, s)-Nets in Base 4

Best Known (164−87, 164, s)-Nets in Base 4

(164−87, 164, 104)-Net over F₄ — Constructive and digital

Digital (77, 164, 104)-net over F₄, using

t-expansion [i] based on digital (73, 164, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(164−87, 164, 112)-Net over F₄ — Digital

Digital (77, 164, 112)-net over F₄, using

t-expansion [i] based on digital (73, 164, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(164−87, 164, 1042)-Net in Base 4 — Upper bound on s

There is no (77, 164, 1043)-net in base 4, because

1 times m-reduction [i] would yield (77, 163, 1043)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 136 739199 908231 417720 628734 221231 633629 078587 979621 100940 825150 239387 879034 065989 532600 523188 215040 > 4¹⁶³ [i]