MinT - Best Known (96−48, 96, s)-Nets in Base 4

Best Known (96−48, 96, s)-Nets in Base 4

(96−48, 96, 57)-Net over F₄ — Constructive and digital

Digital (48, 96, 57)-net over F₄, using

2 times m-reduction [i] based on digital (48, 98, 57)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 35, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (13, 63, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(96−48, 96, 65)-Net in Base 4 — Constructive

(48, 96, 65)-net in base 4, using

6 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]

(96−48, 96, 81)-Net over F₄ — Digital

Digital (48, 96, 81)-net over F₄, using

t-expansion [i] based on digital (46, 96, 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]

(96−48, 96, 817)-Net in Base 4 — Upper bound on s

There is no (48, 96, 818)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6422 239813 650435 834740 766831 124162 547265 615774 919989 221404 > 4⁹⁶ [i]