MinT - Best Known (252−165, 252, s)-Nets in Base 4

Best Known (252−165, 252, s)-Nets in Base 4

(252−165, 252, 104)-Net over F₄ — Constructive and digital

Digital (87, 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]

(252−165, 252, 129)-Net over F₄ — Digital

Digital (87, 252, 129)-net over F₄, using

t-expansion [i] based on digital (81, 252, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(252−165, 252, 661)-Net in Base 4 — Upper bound on s

There is no (87, 252, 662)-net in base 4, because

1 times m-reduction [i] would yield (87, 251, 662)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 147774 913971 106978 923079 268800 107510 389000 022494 470027 264385 059686 024110 051902 781877 665758 819594 993275 190988 734274 830449 675905 738025 414889 614250 603260 > 4²⁵¹ [i]