MinT - Best Known (37, 37+45, s)-Nets in Base 16

Best Known (37, 37+45, s)-Nets in Base 16

(37, 37+45, 130)-Net over F₁₆ — Constructive and digital

Digital (37, 82, 130)-net over F₁₆, using

5 times m-reduction [i] based on digital (37, 87, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 31, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (6, 56, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(37, 37+45, 177)-Net in Base 16 — Constructive

(37, 82, 177)-net in base 16, using

8 times m-reduction [i] based on (37, 90, 177)-net in base 16, using
- base change [i] based on digital (7, 60, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(37, 37+45, 208)-Net over F₁₆ — Digital

Digital (37, 82, 208)-net over F₁₆, using

net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
  - Drinfeld modules of rank 1 [i]

(37, 37+45, 209)-Net in Base 16

(37, 82, 209)-net in base 16, using

2 times m-reduction [i] based on (37, 84, 209)-net in base 16, using
- base change [i] based on digital (9, 56, 209)-net over F₆₄, using
  - net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
      - a curve from the manYPoints database [i]

(37, 37+45, 16360)-Net in Base 16 — Upper bound on s

There is no (37, 82, 16361)-net in base 16, because

1 times m-reduction [i] would yield (37, 81, 16361)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 201260 522166 451321 962407 008468 784583 325827 306384 300367 492350 689790 805346 271242 968054 199997 488856 > 16⁸¹ [i]