MinT - Best Known (13, 13+19, s)-Nets in Base 4

Best Known (13, 13+19, s)-Nets in Base 4

(13, 13+19, 30)-Net over F₄ — Constructive and digital

Digital (13, 32, 30)-net over F₄, using

net from sequence [i] based on digital (13, 29)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
  - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(13, 13+19, 33)-Net over F₄ — Digital

Digital (13, 32, 33)-net over F₄, using

net from sequence [i] based on digital (13, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
  - Drinfeld modules of rank 1 [i]

(13, 13+19, 156)-Net in Base 4 — Upper bound on s

There is no (13, 32, 157)-net in base 4, because

1 times m-reduction [i] would yield (13, 31, 157)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4 625553 860124 859648 > 4³¹ [i]