MinT - Best Known (163−85, 163, s)-Nets in Base 4

Best Known (163−85, 163, s)-Nets in Base 4

(163−85, 163, 104)-Net over F₄ — Constructive and digital

Digital (78, 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]

(163−85, 163, 112)-Net over F₄ — Digital

Digital (78, 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]

(163−85, 163, 1121)-Net in Base 4 — Upper bound on s

There is no (78, 163, 1122)-net in base 4, because

1 times m-reduction [i] would yield (78, 162, 1122)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 467135 258100 904950 765570 023815 147584 097233 145697 848088 902481 207143 280232 422220 156908 497738 505760 > 4¹⁶² [i]