MinT - Best Known (124−75, 124, s)-Nets in Base 4

Best Known (124−75, 124, s)-Nets in Base 4

(124−75, 124, 66)-Net over F₄ — Constructive and digital

Digital (49, 124, 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]

(124−75, 124, 81)-Net over F₄ — Digital

Digital (49, 124, 81)-net over F₄, using

t-expansion [i] based on digital (46, 124, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(124−75, 124, 460)-Net in Base 4 — Upper bound on s

There is no (49, 124, 461)-net in base 4, because

1 times m-reduction [i] would yield (49, 123, 461)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 118 266762 716918 194239 530583 016282 744060 072446 803704 602436 115128 334057 698984 > 4¹²³ [i]