MinT - Best Known (101−53, 101, s)-Nets in Base 4

Best Known (101−53, 101, s)-Nets in Base 4

(101−53, 101, 56)-Net over F₄ — Constructive and digital

Digital (48, 101, 56)-net over F₄, using

t-expansion [i] based on digital (33, 101, 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]

(101−53, 101, 65)-Net in Base 4 — Constructive

(48, 101, 65)-net in base 4, using

1 times m-reduction [i] based on (48, 102, 65)-net in base 4, using
- base change [i] based on digital (14, 68, 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]

(101−53, 101, 81)-Net over F₄ — Digital

Digital (48, 101, 81)-net over F₄, using

t-expansion [i] based on digital (46, 101, 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]

(101−53, 101, 706)-Net in Base 4 — Upper bound on s

There is no (48, 101, 707)-net in base 4, because

1 times m-reduction [i] would yield (48, 100, 707)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 636486 212045 468558 698071 898059 922651 270767 889500 441346 122936 > 4¹⁰⁰ [i]