MinT - Best Known (36, 109, s)-Nets in Base 16

Best Known (36, 109, s)-Nets in Base 16

(36, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (36, 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]

(36, 109, 120)-Net in Base 16 — Constructive

(36, 109, 120)-net in base 16, using

16 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
- base change [i] based on digital (11, 100, 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]

(36, 109, 193)-Net over F₁₆ — Digital

Digital (36, 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]

(36, 109, 3879)-Net in Base 16 — Upper bound on s

There is no (36, 109, 3880)-net in base 16, because

1 times m-reduction [i] would yield (36, 108, 3880)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11132 014594 288590 265678 450158 132566 133180 770176 238060 782738 345344 316212 643649 302636 093702 030374 847004 182608 594489 513510 221357 121451 > 16¹⁰⁸ [i]