MinT - Best Known (237−165, 237, s)-Nets in Base 4

Best Known (237−165, 237, s)-Nets in Base 4

(237−165, 237, 66)-Net over F₄ — Constructive and digital

Digital (72, 237, 66)-net over F₄, using

t-expansion [i] based on digital (49, 237, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(237−165, 237, 105)-Net over F₄ — Digital

Digital (72, 237, 105)-net over F₄, using

t-expansion [i] based on digital (70, 237, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(237−165, 237, 499)-Net in Base 4 — Upper bound on s

There is no (72, 237, 500)-net in base 4, because

1 times m-reduction [i] would yield (72, 236, 500)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12857 567209 402922 687230 797749 893059 755960 947487 875521 584423 324924 134754 641075 093201 671710 105114 149961 010485 708751 869064 099648 656204 629665 039481 > 4²³⁶ [i]