MinT - Best Known (87−63, 87, s)-Nets in Base 16

Best Known (87−63, 87, s)-Nets in Base 16

(87−63, 87, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 87, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 87, 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]

(87−63, 87, 76)-Net in Base 16 — Constructive

(24, 87, 76)-net in base 16, using

8 times m-reduction [i] based on (24, 95, 76)-net in base 16, using
- base change [i] based on digital (5, 76, 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]

(87−63, 87, 129)-Net over F₁₆ — Digital

Digital (24, 87, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 87, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(87−63, 87, 1795)-Net in Base 16 — Upper bound on s

There is no (24, 87, 1796)-net in base 16, because

1 times m-reduction [i] would yield (24, 86, 1796)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 837341 737745 367488 832594 659204 786040 760592 198112 601351 686220 591162 035225 922230 886211 358879 200707 246016 > 16⁸⁶ [i]