MinT - Best Known (117−54, 117, s)-Nets in Base 16

Best Known (117−54, 117, s)-Nets in Base 16

(117−54, 117, 522)-Net over F₁₆ — Constructive and digital

Digital (63, 117, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (63, 118, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 59, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(117−54, 117, 644)-Net over F₁₆ — Digital

Digital (63, 117, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹¹⁷, 644, F₁₆, 2, 54) (dual of [(644, 2), 1171, 55]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹¹⁶, 644, F₁₆, 2, 54) (dual of [(644, 2), 1172, 55]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹¹², 642, F₁₆, 2, 54) (dual of [(642, 2), 1172, 55]-NRT-code), using
    - extracting embedded OOA [i] based on digital (58, 112, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 56, 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]

(117−54, 117, 120258)-Net in Base 16 — Upper bound on s

There is no (63, 117, 120259)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 762 252815 623245 221351 470677 817123 692148 249434 816266 193880 304716 567571 182395 013174 646495 424838 276107 963169 750679 184361 860083 464951 183666 136896 > 16¹¹⁷ [i]