MinT - Best Known (163−87, 163, s)-Nets in Base 4

Best Known (163−87, 163, s)-Nets in Base 4

(163−87, 163, 104)-Net over F₄ — Constructive and digital

Digital (76, 163, 104)-net over F₄, using

t-expansion [i] based on digital (73, 163, 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]

(163−87, 163, 112)-Net over F₄ — Digital

Digital (76, 163, 112)-net over F₄, using

t-expansion [i] based on digital (73, 163, 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]

(163−87, 163, 1008)-Net in Base 4 — Upper bound on s

There is no (76, 163, 1009)-net in base 4, because

1 times m-reduction [i] would yield (76, 162, 1009)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 442921 253965 112413 500005 786631 131620 478117 952325 393109 250172 776503 557396 790658 130763 952601 318224 > 4¹⁶² [i]