MinT - Best Known (28, 28+27, s)-Nets in Base 4

Best Known (28, 28+27, s)-Nets in Base 4

(28, 28+27, 44)-Net over F₄ — Constructive and digital

Digital (28, 55, 44)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 18, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 37, 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]

(28, 28+27, 45)-Net in Base 4 — Constructive

(28, 55, 45)-net in base 4, using

2 times m-reduction [i] based on (28, 57, 45)-net in base 4, using
- base change [i] based on digital (9, 38, 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]

(28, 28+27, 55)-Net over F₄ — Digital

Digital (28, 55, 55)-net over F₄, using

t-expansion [i] based on digital (26, 55, 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]

(28, 28+27, 588)-Net in Base 4 — Upper bound on s

There is no (28, 55, 589)-net in base 4, because

1 times m-reduction [i] would yield (28, 54, 589)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 329 106209 357584 040831 458297 322836 > 4⁵⁴ [i]