MinT - Best Known (56−24, 56, s)-Nets in Base 16

Best Known (56−24, 56, s)-Nets in Base 16

(56−24, 56, 522)-Net over F₁₆ — Constructive and digital

Digital (32, 56, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 28, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(56−24, 56, 644)-Net over F₁₆ — Digital

Digital (32, 56, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁵⁶, 644, F₁₆, 2, 24) (dual of [(644, 2), 1232, 25]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁵², 642, F₁₆, 2, 24) (dual of [(642, 2), 1232, 25]-NRT-code), using
  - extracting embedded OOA [i] based on digital (28, 52, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 26, 321)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
        sharp bound on the number of rational points on curves with genus 2 [i]

(56−24, 56, 146716)-Net in Base 16 — Upper bound on s

There is no (32, 56, 146717)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 26 961694 130071 742988 741061 933559 703859 426477 690389 533522 190107 963611 > 16⁵⁶ [i]