MinT - Best Known (109−77, 109, s)-Nets in Base 16

Best Known (109−77, 109, s)-Nets in Base 16

(109−77, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 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]

(109−77, 109, 104)-Net in Base 16 — Constructive

(32, 109, 104)-net in base 16, using

6 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
- base change [i] based on digital (9, 92, 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]

(109−77, 109, 168)-Net over F₁₆ — Digital

Digital (32, 109, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 109, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(109−77, 109, 2627)-Net in Base 16 — Upper bound on s

There is no (32, 109, 2628)-net in base 16, because

1 times m-reduction [i] would yield (32, 108, 2628)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11220 470818 625536 948548 073609 214625 603281 696599 075634 049591 599221 029411 940721 904124 647819 385462 744940 512407 392897 297121 542496 765136 > 16¹⁰⁸ [i]