MinT - Best Known (38, 38+55, s)-Nets in Base 16

Best Known (38, 38+55, s)-Nets in Base 16

(38, 38+55, 114)-Net over F₁₆ — Constructive and digital

Digital (38, 93, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 32, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (6, 61, 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]

(38, 38+55, 177)-Net in Base 16 — Constructive

(38, 93, 177)-net in base 16, using

base change [i] based on digital (7, 62, 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]

(38, 38+55, 208)-Net over F₁₆ — Digital

Digital (38, 93, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 93, 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]

(38, 38+55, 9216)-Net in Base 16 — Upper bound on s

There is no (38, 93, 9217)-net in base 16, because

1 times m-reduction [i] would yield (38, 92, 9217)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 602 845544 097843 891290 572990 809399 808197 125679 160705 344051 629618 651899 864112 081159 918989 414540 450229 499182 887936 > 16⁹² [i]