MinT - Best Known (71−31, 71, s)-Nets in Base 16

Best Known (71−31, 71, s)-Nets in Base 16

(71−31, 71, 522)-Net over F₁₆ — Constructive and digital

Digital (40, 71, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (40, 72, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 36, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(71−31, 71, 644)-Net over F₁₆ — Digital

Digital (40, 71, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁷¹, 644, F₁₆, 2, 31) (dual of [(644, 2), 1217, 32]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁷⁰, 644, F₁₆, 2, 31) (dual of [(644, 2), 1218, 32]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁶⁶, 642, F₁₆, 2, 31) (dual of [(642, 2), 1218, 32]-NRT-code), using
    - extracting embedded OOA [i] based on digital (35, 66, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 33, 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]

(71−31, 71, 178189)-Net in Base 16 — Upper bound on s

There is no (40, 71, 178190)-net in base 16, because

1 times m-reduction [i] would yield (40, 70, 178190)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 942726 448804 307192 670055 898406 753929 253218 066157 914191 458083 699330 476297 800977 093376 > 16⁷⁰ [i]