MinT - Best Known (70−42, 70, s)-Nets in Base 16

Best Known (70−42, 70, s)-Nets in Base 16

(70−42, 70, 89)-Net over F₁₆ — Constructive and digital

Digital (28, 70, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 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, 48, 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]

(70−42, 70, 120)-Net in Base 16 — Constructive

(28, 70, 120)-net in base 16, using

15 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
- base change [i] based on digital (11, 68, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(70−42, 70, 156)-Net over F₁₆ — Digital

Digital (28, 70, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 70, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(70−42, 70, 5960)-Net in Base 16 — Upper bound on s

There is no (28, 70, 5961)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 943974 885683 522204 247362 933384 341552 692900 448257 757284 907471 410786 570729 709385 541216 > 16⁷⁰ [i]