MinT - Best Known (80−45, 80, s)-Nets in Base 4

Best Known (80−45, 80, s)-Nets in Base 4

(80−45, 80, 56)-Net over F₄ — Constructive and digital

Digital (35, 80, 56)-net over F₄, using

t-expansion [i] based on digital (33, 80, 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]

(80−45, 80, 65)-Net over F₄ — Digital

Digital (35, 80, 65)-net over F₄, using

t-expansion [i] based on digital (33, 80, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(80−45, 80, 420)-Net in Base 4 — Upper bound on s

There is no (35, 80, 421)-net in base 4, because

1 times m-reduction [i] would yield (35, 79, 421)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 371979 500597 434499 172883 611261 121462 857948 242896 > 4⁷⁹ [i]