MinT - Best Known (13−10, 13, s)-Nets in Base 128

Best Known (13−10, 13, s)-Nets in Base 128

(13−10, 13, 192)-Net over F₁₂₈ — Constructive and digital

Digital (3, 13, 192)-net over F₁₂₈, using

net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
  - a function field by SÃ©mirat [i]

(13−10, 13, 258)-Net in Base 128 — Constructive

(3, 13, 258)-net in base 128, using

3 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
- base change [i] based on digital (1, 14, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(13−10, 13, 289)-Net in Base 128

(3, 13, 289)-net in base 128, using

3 times m-reduction [i] based on (3, 16, 289)-net in base 128, using
- base change [i] based on digital (1, 14, 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]

(13−10, 13, 6174)-Net in Base 128 — Upper bound on s

There is no (3, 13, 6175)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2475 934458 850772 674765 330646 > 128¹³ [i]