MinT - Best Known (65, 65+56, s)-Nets in Base 16

Best Known (65, 65+56, s)-Nets in Base 16

(65, 65+56, 522)-Net over F₁₆ — Constructive and digital

Digital (65, 121, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (65, 122, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 61, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(65, 65+56, 644)-Net over F₁₆ — Digital

Digital (65, 121, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹²¹, 644, F₁₆, 2, 56) (dual of [(644, 2), 1167, 57]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹²⁰, 644, F₁₆, 2, 56) (dual of [(644, 2), 1168, 57]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹¹⁶, 642, F₁₆, 2, 56) (dual of [(642, 2), 1168, 57]-NRT-code), using
    - extracting embedded OOA [i] based on digital (60, 116, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 58, 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, 65+56, 120330)-Net in Base 16 — Upper bound on s

There is no (65, 121, 120331)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 950278 086344 692331 978676 154402 963407 111737 894571 179735 830012 914015 949981 173113 820185 614067 910404 813836 144054 590977 336897 233787 705315 082760 269696 > 16¹²¹ [i]