MinT - Best Known (160−85, 160, s)-Nets in Base 4

Best Known (160−85, 160, s)-Nets in Base 4

(160−85, 160, 104)-Net over F₄ — Constructive and digital

Digital (75, 160, 104)-net over F₄, using

t-expansion [i] based on digital (73, 160, 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]

(160−85, 160, 112)-Net over F₄ — Digital

Digital (75, 160, 112)-net over F₄, using

t-expansion [i] based on digital (73, 160, 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]

(160−85, 160, 1012)-Net in Base 4 — Upper bound on s

There is no (75, 160, 1013)-net in base 4, because

1 times m-reduction [i] would yield (75, 159, 1013)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 537560 940091 545286 135447 926887 417146 969721 702204 152708 126299 510460 953694 045834 430003 543111 429860 > 4¹⁵⁹ [i]