MinT - Best Known (105−71, 105, s)-Nets in Base 16

Best Known (105−71, 105, s)-Nets in Base 16

(105−71, 105, 65)-Net over F₁₆ — Constructive and digital

Digital (34, 105, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 105, 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]

(105−71, 105, 120)-Net in Base 16 — Constructive

(34, 105, 120)-net in base 16, using

10 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
- base change [i] based on digital (11, 92, 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]

(105−71, 105, 193)-Net over F₁₆ — Digital

Digital (34, 105, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 105, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(105−71, 105, 3489)-Net in Base 16 — Upper bound on s

There is no (34, 105, 3490)-net in base 16, because

1 times m-reduction [i] would yield (34, 104, 3490)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 170493 641567 214665 436279 494406 070404 766644 479322 622537 903194 778299 684149 932297 191401 793910 647862 170891 795843 985263 308554 125376 > 16¹⁰⁴ [i]