MinT - Best Known (115−82, 115, s)-Nets in Base 16

Best Known (115−82, 115, s)-Nets in Base 16

(115−82, 115, 65)-Net over F₁₆ — Constructive and digital

Digital (33, 115, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 115, 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]

(115−82, 115, 104)-Net in Base 16 — Constructive

(33, 115, 104)-net in base 16, using

5 times m-reduction [i] based on (33, 120, 104)-net in base 16, using
- base change [i] based on digital (9, 96, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(115−82, 115, 193)-Net over F₁₆ — Digital

Digital (33, 115, 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]

(115−82, 115, 2543)-Net in Base 16 — Upper bound on s

There is no (33, 115, 2544)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 017015 688165 200997 177477 555396 941652 160953 662072 049170 283913 526216 076272 866962 980525 493211 520447 117294 084596 499788 088618 991479 478898 219561 > 16¹¹⁵ [i]