MinT - Best Known (48, 48+51, s)-Nets in Base 4

Best Known (48, 48+51, s)-Nets in Base 4

(48, 48+51, 57)-Net over F₄ — Constructive and digital

Digital (48, 99, 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, 64, 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]

(48, 48+51, 65)-Net in Base 4 — Constructive

(48, 99, 65)-net in base 4, using

3 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]

(48, 48+51, 81)-Net over F₄ — Digital

Digital (48, 99, 81)-net over F₄, using

t-expansion [i] based on digital (46, 99, 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]

(48, 48+51, 757)-Net in Base 4 — Upper bound on s

There is no (48, 99, 758)-net in base 4, because

1 times m-reduction [i] would yield (48, 98, 758)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 103206 736870 288777 655680 298806 972528 446388 771397 839481 500804 > 4⁹⁸ [i]