MinT - Best Known (119−55, 119, s)-Nets in Base 16

Best Known (119−55, 119, s)-Nets in Base 16

(119−55, 119, 522)-Net over F₁₆ — Constructive and digital

Digital (64, 119, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (64, 120, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 60, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(119−55, 119, 644)-Net over F₁₆ — Digital

Digital (64, 119, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹¹⁹, 644, F₁₆, 2, 55) (dual of [(644, 2), 1169, 56]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹¹⁸, 644, F₁₆, 2, 55) (dual of [(644, 2), 1170, 56]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹¹⁴, 642, F₁₆, 2, 55) (dual of [(642, 2), 1170, 56]-NRT-code), using
    - extracting embedded OOA [i] based on digital (59, 114, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 57, 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]

(119−55, 119, 133265)-Net in Base 16 — Upper bound on s

There is no (64, 119, 133266)-net in base 16, because

1 times m-reduction [i] would yield (64, 118, 133266)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12195 682805 486255 327661 315002 602227 064645 713763 498421 697983 214690 219785 100160 136590 014622 438799 816327 672185 058455 092587 547881 134362 679399 291456 > 16¹¹⁸ [i]