MinT - Best Known (219−115, 219, s)-Nets in Base 4

Best Known (219−115, 219, s)-Nets in Base 4

(219−115, 219, 104)-Net over F₄ — Constructive and digital

Digital (104, 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]

(219−115, 219, 144)-Net over F₄ — Digital

Digital (104, 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]

(219−115, 219, 1430)-Net in Base 4 — Upper bound on s

There is no (104, 219, 1431)-net in base 4, because

1 times m-reduction [i] would yield (104, 218, 1431)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177907 708535 241444 225995 911232 946044 476575 890427 255533 198862 902708 346503 450286 863321 136949 757359 560454 720360 709936 027855 080128 084256 > 4²¹⁸ [i]