MinT - Best Known (20−12, 20, s)-Nets in Base 16

Best Known (20−12, 20, s)-Nets in Base 16

(20−12, 20, 65)-Net over F₁₆ — Constructive and digital

Digital (8, 20, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 20, 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]

(20−12, 20, 66)-Net over F₁₆ — Digital

Digital (8, 20, 66)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16²⁰, 66, F₁₆, 2, 12) (dual of [(66, 2), 112, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(16¹⁸, 65, F₁₆, 2, 12) (dual of [(65, 2), 112, 13]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,117P) [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(20−12, 20, 80)-Net in Base 16 — Constructive

(8, 20, 80)-net in base 16, using

1 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
- base change [i] based on digital (1, 14, 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]

(20−12, 20, 81)-Net in Base 16

(8, 20, 81)-net in base 16, using

1 times m-reduction [i] based on (8, 21, 81)-net in base 16, using
- base change [i] based on digital (1, 14, 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]

(20−12, 20, 2057)-Net in Base 16 — Upper bound on s

There is no (8, 20, 2058)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 212146 256864 075952 328596 > 16²⁰ [i]