MinT - Best Known (66, 66+57, s)-Nets in Base 16

Best Known (66, 66+57, s)-Nets in Base 16

(66, 66+57, 522)-Net over F₁₆ — Constructive and digital

Digital (66, 123, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (66, 124, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 62, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(66, 66+57, 644)-Net over F₁₆ — Digital

Digital (66, 123, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹²³, 644, F₁₆, 2, 57) (dual of [(644, 2), 1165, 58]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹²², 644, F₁₆, 2, 57) (dual of [(644, 2), 1166, 58]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹¹⁸, 642, F₁₆, 2, 57) (dual of [(642, 2), 1166, 58]-NRT-code), using
    - extracting embedded OOA [i] based on digital (61, 118, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 59, 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]

(66, 66+57, 132857)-Net in Base 16 — Upper bound on s

There is no (66, 123, 132858)-net in base 16, because

1 times m-reduction [i] would yield (66, 122, 132858)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 799 239028 790262 747177 049363 699407 351452 783174 861794 645811 354035 198557 249060 628785 777960 361029 517617 948946 829908 996049 407268 195976 821636 569484 606536 > 16¹²² [i]