MinT - Best Known (72, 72+75, s)-Nets in Base 4

Best Known (72, 72+75, s)-Nets in Base 4

(72, 72+75, 66)-Net over F₄ — Constructive and digital

Digital (72, 147, 66)-net over F₄, using

t-expansion [i] based on digital (49, 147, 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]

(72, 72+75, 105)-Net over F₄ — Digital

Digital (72, 147, 105)-net over F₄, using

t-expansion [i] based on digital (70, 147, 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]

(72, 72+75, 1130)-Net in Base 4 — Upper bound on s

There is no (72, 147, 1131)-net in base 4, because

1 times m-reduction [i] would yield (72, 146, 1131)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8203 605906 864652 662763 467686 922379 053400 955272 095450 521044 975411 831661 923722 924990 895060 > 4¹⁴⁶ [i]