MinT - Best Known (42, 42+41, s)-Nets in Base 4

Best Known (42, 42+41, s)-Nets in Base 4

(42, 42+41, 56)-Net over F₄ — Constructive and digital

Digital (42, 83, 56)-net over F₄, using

t-expansion [i] based on digital (33, 83, 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]

(42, 42+41, 65)-Net in Base 4 — Constructive

(42, 83, 65)-net in base 4, using

1 times m-reduction [i] based on (42, 84, 65)-net in base 4, using
- base change [i] based on digital (14, 56, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(42, 42+41, 75)-Net over F₄ — Digital

Digital (42, 83, 75)-net over F₄, using

t-expansion [i] based on digital (40, 83, 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]

(42, 42+41, 797)-Net in Base 4 — Upper bound on s

There is no (42, 83, 798)-net in base 4, because

1 times m-reduction [i] would yield (42, 82, 798)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23 402137 234906 735177 403063 214257 212747 005760 089181 > 4⁸² [i]