MinT - Best Known (56−22, 56, s)-Nets in Base 8

Best Known (56−22, 56, s)-Nets in Base 8

(56−22, 56, 256)-Net over F₈ — Constructive and digital

Digital (34, 56, 256)-net over F₈, using

2 times m-reduction [i] based on digital (34, 58, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 29, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(56−22, 56, 300)-Net in Base 8 — Constructive

(34, 56, 300)-net in base 8, using

t-expansion [i] based on (33, 56, 300)-net in base 8, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
  - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(56−22, 56, 328)-Net over F₈ — Digital

Digital (34, 56, 328)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−22, 56, 344)-Net in Base 8

(34, 56, 344)-net in base 8, using

trace code for nets [i] based on (6, 28, 172)-net in base 64, using
- base change [i] based on digital (2, 24, 172)-net over F₁₂₈, using
  - net from sequence [i] based on digital (2, 171)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(56−22, 56, 27756)-Net in Base 8 — Upper bound on s

There is no (34, 56, 27757)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 374 240931 463237 445646 736465 623785 388947 347046 985140 > 8⁵⁶ [i]