MinT - Best Known (216−117, 216, s)-Nets in Base 4

Best Known (216−117, 216, s)-Nets in Base 4

(216−117, 216, 104)-Net over F₄ — Constructive and digital

Digital (99, 216, 104)-net over F₄, using

t-expansion [i] based on digital (73, 216, 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]

(216−117, 216, 144)-Net over F₄ — Digital

Digital (99, 216, 144)-net over F₄, using

t-expansion [i] based on digital (91, 216, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(216−117, 216, 1228)-Net in Base 4 — Upper bound on s

There is no (99, 216, 1229)-net in base 4, because

1 times m-reduction [i] would yield (99, 215, 1229)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2782 185730 393117 316517 615740 260525 005345 782284 869836 279431 385828 170219 006060 374941 758148 310480 875598 711784 811492 985398 287177 958488 > 4²¹⁵ [i]