MinT - Best Known (219−117, 219, s)-Nets in Base 4

Best Known (219−117, 219, s)-Nets in Base 4

(219−117, 219, 104)-Net over F₄ — Constructive and digital

Digital (102, 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]

(219−117, 219, 144)-Net over F₄ — Digital

Digital (102, 219, 144)-net over F₄, using

t-expansion [i] based on digital (91, 219, 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]

(219−117, 219, 1323)-Net in Base 4 — Upper bound on s

There is no (102, 219, 1324)-net in base 4, because

1 times m-reduction [i] would yield (102, 218, 1324)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 179406 723514 220281 335697 417858 559473 340610 953204 866492 352918 019456 539758 355072 408303 122392 680589 889925 584962 288789 601344 727544 793616 > 4²¹⁸ [i]