MinT - Best Known (108−75, 108, s)-Nets in Base 4

Best Known (108−75, 108, s)-Nets in Base 4

(108−75, 108, 56)-Net over F₄ — Constructive and digital

Digital (33, 108, 56)-net over F₄, using

net from sequence [i] based on digital (33, 55)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
  - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(108−75, 108, 65)-Net over F₄ — Digital

Digital (33, 108, 65)-net over F₄, using

net from sequence [i] based on digital (33, 64)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
  - Drinfeld modules of rank 1 [i]

(108−75, 108, 239)-Net in Base 4 — Upper bound on s

There is no (33, 108, 240)-net in base 4, because

1 times m-reduction [i] would yield (33, 107, 240)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 26373 350297 026808 744834 411883 171143 109076 171014 911681 675994 241963 > 4¹⁰⁷ [i]