MinT - Best Known (62, 62+53, s)-Nets in Base 16

Best Known (62, 62+53, s)-Nets in Base 16

(62, 62+53, 522)-Net over F₁₆ — Constructive and digital

Digital (62, 115, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (62, 116, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 58, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(62, 62+53, 644)-Net over F₁₆ — Digital

Digital (62, 115, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹¹⁵, 644, F₁₆, 2, 53) (dual of [(644, 2), 1173, 54]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹¹⁴, 644, F₁₆, 2, 53) (dual of [(644, 2), 1174, 54]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹¹⁰, 642, F₁₆, 2, 53) (dual of [(642, 2), 1174, 54]-NRT-code), using
    - extracting embedded OOA [i] based on digital (57, 110, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 55, 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]

(62, 62+53, 133893)-Net in Base 16 — Upper bound on s

There is no (62, 115, 133894)-net in base 16, because

1 times m-reduction [i] would yield (62, 114, 133894)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 186098 540150 961429 643383 816952 985506 478611 716942 988272 753902 747839 463280 891378 012079 020272 293541 565632 375406 401150 790040 275327 057363 065536 > 16¹¹⁴ [i]