MinT - Best Known (26, 26+19, s)-Nets in Base 16

Best Known (26, 26+19, s)-Nets in Base 16

(26, 26+19, 520)-Net over F₁₆ — Constructive and digital

Digital (26, 45, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (26, 46, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 23, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(26, 26+19, 643)-Net over F₁₆ — Digital

Digital (26, 45, 643)-net over F₁₆, using

16¹ times duplication [i] based on digital (25, 44, 643)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁴, 643, F₁₆, 2, 19) (dual of [(643, 2), 1242, 20]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(16⁴², 642, F₁₆, 2, 19) (dual of [(642, 2), 1242, 20]-NRT-code), using
    - extracting embedded OOA [i] based on digital (23, 42, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 21, 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]

(26, 26+19, 213039)-Net in Base 16 — Upper bound on s

There is no (26, 45, 213040)-net in base 16, because

1 times m-reduction [i] would yield (26, 44, 213040)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 95784 023137 074269 407790 142944 489498 941841 906246 526151 > 16⁴⁴ [i]