MinT - Best Known (58−25, 58, s)-Nets in Base 16

Best Known (58−25, 58, s)-Nets in Base 16

(58−25, 58, 522)-Net over F₁₆ — Constructive and digital

Digital (33, 58, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 29, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(58−25, 58, 644)-Net over F₁₆ — Digital

Digital (33, 58, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁵⁸, 644, F₁₆, 2, 25) (dual of [(644, 2), 1230, 26]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁵⁴, 642, F₁₆, 2, 25) (dual of [(642, 2), 1230, 26]-NRT-code), using
  - extracting embedded OOA [i] based on digital (29, 54, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 27, 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]

(58−25, 58, 184852)-Net in Base 16 — Upper bound on s

There is no (33, 58, 184853)-net in base 16, because

1 times m-reduction [i] would yield (33, 57, 184853)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 431 373828 708903 308954 716534 954604 501171 036133 598769 480301 391583 213266 > 16⁵⁷ [i]