MinT - Best Known (219−123, 219, s)-Nets in Base 4

Best Known (219−123, 219, s)-Nets in Base 4

(219−123, 219, 104)-Net over F₄ — Constructive and digital

Digital (96, 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]

(219−123, 219, 144)-Net over F₄ — Digital

Digital (96, 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]

(219−123, 219, 1064)-Net in Base 4 — Upper bound on s

There is no (96, 219, 1065)-net in base 4, because

1 times m-reduction [i] would yield (96, 218, 1065)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186389 213182 179889 389543 795691 326471 978967 878489 417424 226053 111333 102518 413582 057843 884919 985806 032505 806825 541027 717611 019222 984704 > 4²¹⁸ [i]