MinT - Best Known (49−23, 49, s)-Nets in Base 16

Best Known (49−23, 49, s)-Nets in Base 16

(49−23, 49, 516)-Net over F₁₆ — Constructive and digital

Digital (26, 49, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (26, 50, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 25, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(49−23, 49, 578)-Net over F₁₆ — Digital

Digital (26, 49, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (26, 50, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 25, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(49−23, 49, 58779)-Net in Base 16 — Upper bound on s

There is no (26, 49, 58780)-net in base 16, because

1 times m-reduction [i] would yield (26, 48, 58780)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6278 262384 980538 882642 457078 644635 676015 709278 091854 556201 > 16⁴⁸ [i]