MinT - Best Known (89−40, 89, s)-Nets in Base 16

Best Known (89−40, 89, s)-Nets in Base 16

(89−40, 89, 522)-Net over F₁₆ — Constructive and digital

Digital (49, 89, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (49, 90, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(89−40, 89, 643)-Net over F₁₆ — Digital

Digital (49, 89, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁸⁹, 643, F₁₆, 2, 40) (dual of [(643, 2), 1197, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16⁸⁹, 644, F₁₆, 2, 40) (dual of [(644, 2), 1199, 41]-NRT-code), using
  - 16¹ times duplication [i] based on linear OOA(16⁸⁸, 644, F₁₆, 2, 40) (dual of [(644, 2), 1200, 41]-NRT-code), using
    - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁸⁴, 642, F₁₆, 2, 40) (dual of [(642, 2), 1200, 41]-NRT-code), using
      - extracting embedded OOA [i] based on digital (44, 84, 642)-net over F₁₆, using
        trace code for nets [i] based on digital (2, 42, 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]

(89−40, 89, 126331)-Net in Base 16 — Upper bound on s

There is no (49, 89, 126332)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146795 515643 546174 636916 408678 993580 266932 296995 161347 402837 589079 228269 316861 827851 510403 988702 047477 502226 > 16⁸⁹ [i]