MinT - Best Known (65−28, 65, s)-Nets in Base 16

Best Known (65−28, 65, s)-Nets in Base 16

(65−28, 65, 522)-Net over F₁₆ — Constructive and digital

Digital (37, 65, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (37, 66, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 33, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(65−28, 65, 644)-Net over F₁₆ — Digital

Digital (37, 65, 644)-net over F₁₆, using

16¹ times duplication [i] based on digital (36, 64, 644)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁶⁴, 644, F₁₆, 2, 28) (dual of [(644, 2), 1224, 29]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁶⁰, 642, F₁₆, 2, 28) (dual of [(642, 2), 1224, 29]-NRT-code), using
    - extracting embedded OOA [i] based on digital (32, 60, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 30, 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]

(65−28, 65, 157001)-Net in Base 16 — Upper bound on s

There is no (37, 65, 157002)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 852805 015140 922847 609482 213362 714826 283160 464477 447300 674731 606400 116404 952096 > 16⁶⁵ [i]