MinT - Best Known (31, 31+49, s)-Nets in Base 16

Best Known (31, 31+49, s)-Nets in Base 16

(31, 31+49, 89)-Net over F₁₆ — Constructive and digital

Digital (31, 80, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 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, 55, 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]

(31, 31+49, 120)-Net in Base 16 — Constructive

(31, 80, 120)-net in base 16, using

20 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
- base change [i] based on digital (11, 80, 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]

(31, 31+49, 168)-Net over F₁₆ — Digital

Digital (31, 80, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(31, 31+49, 5996)-Net in Base 16 — Upper bound on s

There is no (31, 80, 5997)-net in base 16, because

1 times m-reduction [i] would yield (31, 79, 5997)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133854 425235 580028 276475 241115 753257 310239 702733 167317 187012 493694 258112 241830 757108 464099 467346 > 16⁷⁹ [i]