MinT - Best Known (108−78, 108, s)-Nets in Base 16

Best Known (108−78, 108, s)-Nets in Base 16

(108−78, 108, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 108, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 108, 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]

(108−78, 108, 98)-Net in Base 16 — Constructive

(30, 108, 98)-net in base 16, using

7 times m-reduction [i] based on (30, 115, 98)-net in base 16, using
- base change [i] based on digital (7, 92, 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]

(108−78, 108, 162)-Net over F₁₆ — Digital

Digital (30, 108, 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]

(108−78, 108, 2195)-Net in Base 16 — Upper bound on s

There is no (30, 108, 2196)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11143 479980 132393 693278 420800 420603 336006 250763 232128 465020 450797 397783 545320 559020 932426 160720 840491 645096 587740 067031 551921 189136 > 16¹⁰⁸ [i]