MinT - Best Known (28−12, 28, s)-Nets in Base 8

Best Known (28−12, 28, s)-Nets in Base 8

(28−12, 28, 160)-Net over F₈ — Constructive and digital

Digital (16, 28, 160)-net over F₈, using

2 times m-reduction [i] based on digital (16, 30, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 15, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(28−12, 28, 194)-Net over F₈ — Digital

Digital (16, 28, 194)-net over F₈, using

trace code for nets [i] based on digital (2, 14, 97)-net over F₆₄, using
- net from sequence [i] based on digital (2, 96)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 2 and N(F) ≥ 97, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(28−12, 28, 258)-Net in Base 8 — Constructive

(16, 28, 258)-net in base 8, using

trace code for nets [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(28−12, 28, 7003)-Net in Base 8 — Upper bound on s

There is no (16, 28, 7004)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 19 345548 709461 354541 298362 > 8²⁸ [i]