MinT - Best Known (252−159, 252, s)-Nets in Base 4

Best Known (252−159, 252, s)-Nets in Base 4

(252−159, 252, 104)-Net over F₄ — Constructive and digital

Digital (93, 252, 104)-net over F₄, using

t-expansion [i] based on digital (73, 252, 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]

(252−159, 252, 144)-Net over F₄ — Digital

Digital (93, 252, 144)-net over F₄, using

t-expansion [i] based on digital (91, 252, 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]

(252−159, 252, 760)-Net in Base 4 — Upper bound on s

There is no (93, 252, 761)-net in base 4, because

1 times m-reduction [i] would yield (93, 251, 761)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 103495 042889 769172 175730 813070 922430 206193 560603 844274 359146 138340 250200 482865 612115 612169 728074 562285 433515 766959 651048 006065 847031 832158 508325 169424 > 4²⁵¹ [i]