MinT - Best Known (159−93, 159, s)-Nets in Base 4

Best Known (159−93, 159, s)-Nets in Base 4

(159−93, 159, 66)-Net over F₄ — Constructive and digital

Digital (66, 159, 66)-net over F₄, using

t-expansion [i] based on digital (49, 159, 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]

(159−93, 159, 99)-Net over F₄ — Digital

Digital (66, 159, 99)-net over F₄, using

t-expansion [i] based on digital (61, 159, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(159−93, 159, 664)-Net in Base 4 — Upper bound on s

There is no (66, 159, 665)-net in base 4, because

1 times m-reduction [i] would yield (66, 158, 665)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 138238 775998 594440 535993 119704 526861 888880 204059 813496 604047 588220 796586 513483 021071 826815 657952 > 4¹⁵⁸ [i]