MinT - Best Known (72, 72+113, s)-Nets in Base 4

Best Known (72, 72+113, s)-Nets in Base 4

(72, 72+113, 66)-Net over F₄ — Constructive and digital

Digital (72, 185, 66)-net over F₄, using

t-expansion [i] based on digital (49, 185, 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+113, 105)-Net over F₄ — Digital

Digital (72, 185, 105)-net over F₄, using

t-expansion [i] based on digital (70, 185, 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+113, 643)-Net in Base 4 — Upper bound on s

There is no (72, 185, 644)-net in base 4, because

1 times m-reduction [i] would yield (72, 184, 644)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 642 774698 754476 902254 414158 764466 697460 284415 339993 901524 097370 682495 469882 794038 171877 973259 343202 364500 524624 > 4¹⁸⁴ [i]