MinT - Best Known (81, 216, s)-Nets in Base 4

Best Known (81, 216, s)-Nets in Base 4

(81, 216, 104)-Net over F₄ — Constructive and digital

Digital (81, 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]

(81, 216, 129)-Net over F₄ — Digital

Digital (81, 216, 129)-net over F₄, using

net from sequence [i] based on digital (81, 128)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
  - Drinfeld modules of rank 1 [i]

(81, 216, 681)-Net in Base 4 — Upper bound on s

There is no (81, 216, 682)-net in base 4, because

1 times m-reduction [i] would yield (81, 215, 682)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3007 957911 840534 515760 979757 505310 581817 146038 627314 949464 639329 920319 706005 413161 449262 314915 317404 587536 999562 213875 599225 702840 > 4²¹⁵ [i]