MinT - Best Known (82, 227, s)-Nets in Base 4

Best Known (82, 227, s)-Nets in Base 4

(82, 227, 104)-Net over F₄ — Constructive and digital

Digital (82, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(82, 227, 129)-Net over F₄ — Digital

Digital (82, 227, 129)-net over F₄, using

t-expansion [i] based on digital (81, 227, 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]

(82, 227, 657)-Net in Base 4 — Upper bound on s

There is no (82, 227, 658)-net in base 4, because

1 times m-reduction [i] would yield (82, 226, 658)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12763 468293 810222 308482 477804 494313 459493 629265 404667 530530 185732 887707 031483 141695 647482 988279 811540 968715 597911 575367 503924 492727 824180 > 4²²⁶ [i]