MinT - Best Known (24, 24+18, s)-Nets in Base 16

Best Known (24, 24+18, s)-Nets in Base 16

(24, 24+18, 520)-Net over F₁₆ — Constructive and digital

Digital (24, 42, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 21, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(24, 24+18, 643)-Net over F₁₆ — Digital

Digital (24, 42, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴², 643, F₁₆, 2, 18) (dual of [(643, 2), 1244, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(16⁴⁰, 642, F₁₆, 2, 18) (dual of [(642, 2), 1244, 19]-NRT-code), using
  - extracting embedded OOA [i] based on digital (22, 40, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 20, 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]

(24, 24+18, 115045)-Net in Base 16 — Upper bound on s

There is no (24, 42, 115046)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 374 160802 148838 276075 206288 518567 327690 308786 816436 > 16⁴² [i]