MinT - Best Known (113−81, 113, s)-Nets in Base 16

Best Known (113−81, 113, s)-Nets in Base 16

(113−81, 113, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 113, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 113, 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]

(113−81, 113, 104)-Net in Base 16 — Constructive

(32, 113, 104)-net in base 16, using

2 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
- base change [i] based on digital (9, 92, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(113−81, 113, 168)-Net over F₁₆ — Digital

Digital (32, 113, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 113, 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]

(113−81, 113, 2450)-Net in Base 16 — Upper bound on s

There is no (32, 113, 2451)-net in base 16, because

1 times m-reduction [i] would yield (32, 112, 2451)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 727 436038 835741 310291 554637 460846 831876 692443 009761 325778 920826 496243 801891 984110 972841 596578 357967 732003 877934 617464 140569 487120 037976 > 16¹¹² [i]