MinT - Best Known (216−111, 216, s)-Nets in Base 4

Best Known (216−111, 216, s)-Nets in Base 4

(216−111, 216, 130)-Net over F₄ — Constructive and digital

Digital (105, 216, 130)-net over F₄, using

net from sequence [i] based on digital (105, 129)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
  - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(216−111, 216, 144)-Net over F₄ — Digital

Digital (105, 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]

(216−111, 216, 1560)-Net in Base 4 — Upper bound on s

There is no (105, 216, 1561)-net in base 4, because

1 times m-reduction [i] would yield (105, 215, 1561)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2829 957970 927020 532797 410115 338159 998033 590365 196403 861389 301250 785267 287609 394642 452259 785622 080094 353825 223282 542436 229052 565120 > 4²¹⁵ [i]