MinT - Best Known (219−144, 219, s)-Nets in Base 4

Best Known (219−144, 219, s)-Nets in Base 4

(219−144, 219, 104)-Net over F₄ — Constructive and digital

Digital (75, 219, 104)-net over F₄, using

t-expansion [i] based on digital (73, 219, 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]

(219−144, 219, 112)-Net over F₄ — Digital

Digital (75, 219, 112)-net over F₄, using

t-expansion [i] based on digital (73, 219, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(219−144, 219, 567)-Net in Base 4 — Upper bound on s

There is no (75, 219, 568)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 778390 939377 259302 917034 019103 584217 547756 596623 658255 803761 922437 472542 645130 441175 269430 024628 211701 308838 700093 754261 742885 319216 > 4²¹⁹ [i]