MinT - Best Known (113−85, 113, s)-Nets in Base 16

Best Known (113−85, 113, s)-Nets in Base 16

(113−85, 113, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 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]

(113−85, 113, 76)-Net in Base 16 — Constructive

(28, 113, 76)-net in base 16, using

2 times m-reduction [i] based on (28, 115, 76)-net in base 16, using
- base change [i] based on digital (5, 92, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(113−85, 113, 156)-Net over F₁₆ — Digital

Digital (28, 113, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 113, 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]

(113−85, 113, 1766)-Net in Base 16 — Upper bound on s

There is no (28, 113, 1767)-net in base 16, because

1 times m-reduction [i] would yield (28, 112, 1767)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 741 780024 678820 644100 048763 338364 820142 661368 832498 798245 731933 464595 572012 391518 085589 190557 459263 695435 057682 799001 623709 343964 458136 > 16¹¹² [i]