MinT - Best Known (23−14, 23, s)-Nets in Base 16

Best Known (23−14, 23, s)-Nets in Base 16

(23−14, 23, 65)-Net over F₁₆ — Constructive and digital

Digital (9, 23, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 23, 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]

(23−14, 23, 72)-Net over F₁₆ — Digital

Digital (9, 23, 72)-net over F₁₆, using

net from sequence [i] based on digital (9, 71)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 9 and N(F) ≥ 72, using
  - a curve from the manYPoints database [i]

(23−14, 23, 80)-Net in Base 16 — Constructive

(9, 23, 80)-net in base 16, using

1 times m-reduction [i] based on (9, 24, 80)-net in base 16, using
- base change [i] based on digital (1, 16, 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]

(23−14, 23, 81)-Net in Base 16

(9, 23, 81)-net in base 16, using

1 times m-reduction [i] based on (9, 24, 81)-net in base 16, using
- base change [i] based on digital (1, 16, 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]

(23−14, 23, 2034)-Net in Base 16 — Upper bound on s

There is no (9, 23, 2035)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4958 279695 414011 909955 484176 > 16²³ [i]