MinT - Best Known (49−28, 49, s)-Nets in Base 16

Best Known (49−28, 49, s)-Nets in Base 16

(49−28, 49, 89)-Net over F₁₆ — Constructive and digital

Digital (21, 49, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 34, 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]

(49−28, 49, 129)-Net in Base 16 — Constructive

(21, 49, 129)-net in base 16, using

base change [i] based on digital (0, 28, 129)-net over F₁₂₈, using
- net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
    - the rational function field F₁₂₈(x) [i]
  - Niederreiter sequence [i]

(49−28, 49, 129)-Net over F₁₆ — Digital

Digital (21, 49, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 49, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(49−28, 49, 6596)-Net in Base 16 — Upper bound on s

There is no (21, 49, 6597)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 100592 381503 284921 314231 085751 695017 026120 190690 168830 043296 > 16⁴⁹ [i]