MinT - Best Known (218−133, 218, s)-Nets in Base 4

Best Known (218−133, 218, s)-Nets in Base 4

(218−133, 218, 104)-Net over F₄ — Constructive and digital

Digital (85, 218, 104)-net over F₄, using

t-expansion [i] based on digital (73, 218, 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]

(218−133, 218, 129)-Net over F₄ — Digital

Digital (85, 218, 129)-net over F₄, using

t-expansion [i] based on digital (81, 218, 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]

(218−133, 218, 754)-Net in Base 4 — Upper bound on s

There is no (85, 218, 755)-net in base 4, because

1 times m-reduction [i] would yield (85, 217, 755)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44504 725253 218792 870957 686766 740207 973605 131901 562622 491595 080907 715289 337035 503207 059458 951106 017505 299315 363395 379948 994378 855989 > 4²¹⁷ [i]