MinT - Best Known (7, 18, s)-Nets in Base 16

Best Known (7, 18, s)-Nets in Base 16

(7, 18, 65)-Net over F₁₆ — Constructive and digital

Digital (7, 18, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 18, 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]

(7, 18, 80)-Net in Base 16 — Constructive

(7, 18, 80)-net in base 16, using

base change [i] based on digital (1, 12, 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]

(7, 18, 81)-Net in Base 16

(7, 18, 81)-net in base 16, using

base change [i] based on digital (1, 12, 81)-net over F₆₄, using
- net from sequence [i] based on digital (1, 80)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 81, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(7, 18, 2154)-Net in Base 16 — Upper bound on s

There is no (7, 18, 2155)-net in base 16, because

1 times m-reduction [i] would yield (7, 17, 2155)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 295 706261 348267 257876 > 16¹⁷ [i]