MinT - Best Known (56, 103, s)-Nets in Base 16

Best Known (56, 103, s)-Nets in Base 16

(56, 103, 522)-Net over F₁₆ — Constructive and digital

Digital (56, 103, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 104, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 52, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(56, 103, 644)-Net over F₁₆ — Digital

Digital (56, 103, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹⁰³, 644, F₁₆, 2, 47) (dual of [(644, 2), 1185, 48]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹⁰², 644, F₁₆, 2, 47) (dual of [(644, 2), 1186, 48]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁹⁸, 642, F₁₆, 2, 47) (dual of [(642, 2), 1186, 48]-NRT-code), using
    - extracting embedded OOA [i] based on digital (51, 98, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 49, 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]

(56, 103, 137510)-Net in Base 16 — Upper bound on s

There is no (56, 103, 137511)-net in base 16, because

1 times m-reduction [i] would yield (56, 102, 137511)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 661 113482 207473 151440 654104 801018 865085 321526 658622 978449 163598 233397 420688 426688 474227 257445 321590 997283 970584 074498 756096 > 16¹⁰² [i]