MinT - Best Known (219−145, 219, s)-Nets in Base 4

Best Known (219−145, 219, s)-Nets in Base 4

(219−145, 219, 104)-Net over F₄ — Constructive and digital

Digital (74, 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]

(219−145, 219, 112)-Net over F₄ — Digital

Digital (74, 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]

(219−145, 219, 555)-Net in Base 4 — Upper bound on s

There is no (74, 219, 556)-net in base 4, because

1 times m-reduction [i] would yield (74, 218, 556)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 192070 675438 978959 443689 414933 069382 111966 272435 924090 888521 048514 827589 787794 230877 459095 022911 131198 784383 714878 585001 364852 889192 > 4²¹⁸ [i]