MinT - Best Known (85−38, 85, s)-Nets in Base 16

Best Known (85−38, 85, s)-Nets in Base 16

(85−38, 85, 522)-Net over F₁₆ — Constructive and digital

Digital (47, 85, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (47, 86, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 43, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(85−38, 85, 644)-Net over F₁₆ — Digital

Digital (47, 85, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁸⁵, 644, F₁₆, 2, 38) (dual of [(644, 2), 1203, 39]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁸⁴, 644, F₁₆, 2, 38) (dual of [(644, 2), 1204, 39]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁸⁰, 642, F₁₆, 2, 38) (dual of [(642, 2), 1204, 39]-NRT-code), using
    - extracting embedded OOA [i] based on digital (42, 80, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 40, 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]

(85−38, 85, 128807)-Net in Base 16 — Upper bound on s

There is no (47, 85, 128808)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 239808 348644 186258 167979 164428 524820 943843 906115 051847 504790 928945 941098 120976 791549 870772 980289 286006 > 16⁸⁵ [i]