MinT - Best Known (128−88, 128, s)-Nets in Base 4

Best Known (128−88, 128, s)-Nets in Base 4

(128−88, 128, 56)-Net over F₄ — Constructive and digital

Digital (40, 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]

(128−88, 128, 75)-Net over F₄ — Digital

Digital (40, 128, 75)-net over F₄, using

net from sequence [i] based on digital (40, 74)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
  - Drinfeld modules of rank 1 [i]

(128−88, 128, 289)-Net in Base 4 — Upper bound on s

There is no (40, 128, 290)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 118689 861538 997091 592929 156103 685461 202210 660412 499386 123787 863080 694960 927040 > 4¹²⁸ [i]