MinT - Best Known (117−87, 117, s)-Nets in Base 16

Best Known (117−87, 117, s)-Nets in Base 16

(117−87, 117, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 117, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 117, 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]

(117−87, 117, 76)-Net in Base 16 — Constructive

(30, 117, 76)-net in base 16, using

8 times m-reduction [i] based on (30, 125, 76)-net in base 16, using
- base change [i] based on digital (5, 100, 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]

(117−87, 117, 162)-Net over F₁₆ — Digital

Digital (30, 117, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(117−87, 117, 1970)-Net in Base 16 — Upper bound on s

There is no (30, 117, 1971)-net in base 16, because

1 times m-reduction [i] would yield (30, 116, 1971)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 48 575092 366173 544196 266220 060631 837157 433287 587784 800637 634696 786697 635611 460690 131313 336121 049487 582478 422681 298034 978075 661664 226784 338096 > 16¹¹⁶ [i]