MinT - Best Known (109−74, 109, s)-Nets in Base 16

Best Known (109−74, 109, s)-Nets in Base 16

(109−74, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (35, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−74, 109, 120)-Net in Base 16 — Constructive

(35, 109, 120)-net in base 16, using

11 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
- base change [i] based on digital (11, 96, 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]

(109−74, 109, 193)-Net over F₁₆ — Digital

Digital (35, 109, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 109, 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]

(109−74, 109, 3423)-Net in Base 16 — Upper bound on s

There is no (35, 109, 3424)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 177832 897202 083072 359186 178988 507058 320146 817198 001304 626755 817677 186268 004072 943013 542447 283571 740429 440156 468315 967281 032881 750946 > 16¹⁰⁹ [i]