MinT - Best Known (27, 27+20, s)-Nets in Base 16

Best Known (27, 27+20, s)-Nets in Base 16

(27, 27+20, 520)-Net over F₁₆ — Constructive and digital

Digital (27, 47, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (27, 48, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 24, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(27, 27+20, 643)-Net over F₁₆ — Digital

Digital (27, 47, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁷, 643, F₁₆, 2, 20) (dual of [(643, 2), 1239, 21]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁴⁶, 643, F₁₆, 2, 20) (dual of [(643, 2), 1240, 21]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(16⁴⁴, 642, F₁₆, 2, 20) (dual of [(642, 2), 1240, 21]-NRT-code), using
    - extracting embedded OOA [i] based on digital (24, 44, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 22, 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]

(27, 27+20, 137794)-Net in Base 16 — Upper bound on s

There is no (27, 47, 137795)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 392 325904 278351 599352 801160 025196 954218 423251 062293 892376 > 16⁴⁷ [i]