MinT - Best Known (128−81, 128, s)-Nets in Base 4

Best Known (128−81, 128, s)-Nets in Base 4

(128−81, 128, 56)-Net over F₄ — Constructive and digital

Digital (47, 128, 56)-net over F₄, using

t-expansion [i] based on digital (33, 128, 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]

(128−81, 128, 81)-Net over F₄ — Digital

Digital (47, 128, 81)-net over F₄, using

t-expansion [i] based on digital (46, 128, 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]

(128−81, 128, 396)-Net in Base 4 — Upper bound on s

There is no (47, 128, 397)-net in base 4, because

1 times m-reduction [i] would yield (47, 127, 397)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 29357 865496 860469 023471 345140 645916 557658 692683 254189 596433 932013 332509 327804 > 4¹²⁷ [i]