MinT - Best Known (193−119, 193, s)-Nets in Base 4

Best Known (193−119, 193, s)-Nets in Base 4

(193−119, 193, 104)-Net over F₄ — Constructive and digital

Digital (74, 193, 104)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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]

(193−119, 193, 112)-Net over F₄ — Digital

Digital (74, 193, 112)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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]

(193−119, 193, 645)-Net in Base 4 — Upper bound on s

There is no (74, 193, 646)-net in base 4, because

1 times m-reduction [i] would yield (74, 192, 646)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42 312208 002671 528241 848360 528337 290891 465604 004573 888737 936751 206111 570376 088625 202200 446654 960782 438803 897968 270176 > 4¹⁹² [i]