MinT - Best Known (31, 31+35, s)-Nets in Base 4

Best Known (31, 31+35, s)-Nets in Base 4

(31, 31+35, 42)-Net over F₄ — Constructive and digital

Digital (31, 66, 42)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 42, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄ (see above)

(31, 31+35, 45)-Net in Base 4 — Constructive

(31, 66, 45)-net in base 4, using

base change [i] based on digital (9, 44, 45)-net over F₈, using
- net from sequence [i] based on digital (9, 44)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
    - an explicitly constructive algebraic function field [i]

(31, 31+35, 60)-Net over F₄ — Digital

Digital (31, 66, 60)-net over F₄, using

net from sequence [i] based on digital (31, 59)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 31 and N(F) ≥ 60, using
  - Drinfeld modules of rank 1 [i]

(31, 31+35, 466)-Net in Base 4 — Upper bound on s

There is no (31, 66, 467)-net in base 4, because

1 times m-reduction [i] would yield (31, 65, 467)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1409 475484 151772 515279 192955 698140 145890 > 4⁶⁵ [i]