MinT - Best Known (56−29, 56, s)-Nets in Base 4

Best Known (56−29, 56, s)-Nets in Base 4

(56−29, 56, 41)-Net over F₄ — Constructive and digital

Digital (27, 56, 41)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 39, 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]

(56−29, 56, 42)-Net in Base 4 — Constructive

(27, 56, 42)-net in base 4, using

net from sequence [i] based on (27, 41)-sequence in base 4, using
- base expansion [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(56−29, 56, 55)-Net over F₄ — Digital

Digital (27, 56, 55)-net over F₄, using

t-expansion [i] based on digital (26, 56, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(56−29, 56, 456)-Net in Base 4 — Upper bound on s

There is no (27, 56, 457)-net in base 4, because

1 times m-reduction [i] would yield (27, 55, 457)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1329 119237 578707 184819 838701 541920 > 4⁵⁵ [i]