MinT - Best Known (127−81, 127, s)-Nets in Base 4

Best Known (127−81, 127, s)-Nets in Base 4

(127−81, 127, 56)-Net over F₄ — Constructive and digital

Digital (46, 127, 56)-net over F₄, using

t-expansion [i] based on digital (33, 127, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(127−81, 127, 81)-Net over F₄ — Digital

Digital (46, 127, 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]

(127−81, 127, 382)-Net in Base 4 — Upper bound on s

There is no (46, 127, 383)-net in base 4, because

1 times m-reduction [i] would yield (46, 126, 383)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7761 901593 337877 412762 633584 487369 402656 851754 444248 832871 399800 184917 679637 > 4¹²⁶ [i]