MinT - Best Known (27−16, 27, s)-Nets in Base 16

Best Known (27−16, 27, s)-Nets in Base 16

(27−16, 27, 65)-Net over F₁₆ — Constructive and digital

Digital (11, 27, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 27, 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]

(27−16, 27, 80)-Net in Base 16 — Constructive

(11, 27, 80)-net in base 16, using

3 times m-reduction [i] based on (11, 30, 80)-net in base 16, using
- base change [i] based on digital (1, 20, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(27−16, 27, 81)-Net over F₁₆ — Digital

Digital (11, 27, 81)-net over F₁₆, using

t-expansion [i] based on digital (10, 27, 81)-net over F₁₆, using
- net from sequence [i] based on digital (10, 80)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 10 and N(F) ≥ 81, using
    - a curve from the manYPoints database [i]

(27−16, 27, 97)-Net in Base 16

(11, 27, 97)-net in base 16, using

base change [i] based on digital (2, 18, 97)-net over F₆₄, using
- net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(27−16, 27, 2903)-Net in Base 16 — Upper bound on s

There is no (11, 27, 2904)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 325 089315 217427 063595 759090 597106 > 16²⁷ [i]