MinT - Best Known (102−73, 102, s)-Nets in Base 16

Best Known (102−73, 102, s)-Nets in Base 16

(102−73, 102, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 102, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 102, 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]

(102−73, 102, 98)-Net in Base 16 — Constructive

(29, 102, 98)-net in base 16, using

8 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
- base change [i] based on digital (7, 88, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(102−73, 102, 161)-Net over F₁₆ — Digital

Digital (29, 102, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(102−73, 102, 2254)-Net in Base 16 — Upper bound on s

There is no (29, 102, 2255)-net in base 16, because

1 times m-reduction [i] would yield (29, 101, 2255)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 600232 260334 124432 332096 375793 617571 592032 027716 785309 024039 148644 425065 472393 440315 262042 485475 244412 640802 684151 540826 > 16¹⁰¹ [i]