MinT - Best Known (63−31, 63, s)-Nets in Base 4

Best Known (63−31, 63, s)-Nets in Base 4

(63−31, 63, 48)-Net over F₄ — Constructive and digital

Digital (32, 63, 48)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 22, 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 (10, 41, 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]

(63−31, 63, 60)-Net over F₄ — Digital

Digital (32, 63, 60)-net over F₄, using

t-expansion [i] based on digital (31, 63, 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]

(63−31, 63, 647)-Net in Base 4 — Upper bound on s

There is no (32, 63, 648)-net in base 4, because

1 times m-reduction [i] would yield (32, 62, 648)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21 510381 836041 376797 397788 372088 542576 > 4⁶² [i]