MinT - Best Known (79−35, 79, s)-Nets in Base 16

Best Known (79−35, 79, s)-Nets in Base 16

(79−35, 79, 522)-Net over F₁₆ — Constructive and digital

Digital (44, 79, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 80, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 40, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(79−35, 79, 644)-Net over F₁₆ — Digital

Digital (44, 79, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁷⁹, 644, F₁₆, 2, 35) (dual of [(644, 2), 1209, 36]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁷⁸, 644, F₁₆, 2, 35) (dual of [(644, 2), 1210, 36]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁷⁴, 642, F₁₆, 2, 35) (dual of [(642, 2), 1210, 36]-NRT-code), using
    - extracting embedded OOA [i] based on digital (39, 74, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 37, 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]

(79−35, 79, 160182)-Net in Base 16 — Upper bound on s

There is no (44, 79, 160183)-net in base 16, because

1 times m-reduction [i] would yield (44, 78, 160183)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8344 167793 147715 453510 195093 937022 166444 112542 964356 368984 547860 652439 438869 230710 483674 640766 > 16⁷⁸ [i]