MinT - Best Known (64−28, 64, s)-Nets in Base 16

Best Known (64−28, 64, s)-Nets in Base 16

(64−28, 64, 522)-Net over F₁₆ — Constructive and digital

Digital (36, 64, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 32, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(64−28, 64, 644)-Net over F₁₆ — Digital

Digital (36, 64, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁶⁴, 644, F₁₆, 2, 28) (dual of [(644, 2), 1224, 29]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁶⁰, 642, F₁₆, 2, 28) (dual of [(642, 2), 1224, 29]-NRT-code), using
  - extracting embedded OOA [i] based on digital (32, 60, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 30, 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]

(64−28, 64, 128792)-Net in Base 16 — Upper bound on s

There is no (36, 64, 128793)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115800 866092 967210 610019 568409 028675 044281 893440 682769 105109 925359 923987 865256 > 16⁶⁴ [i]