MinT - Best Known (4, 13, s)-Nets in Base 128

Best Known (4, 13, s)-Nets in Base 128

(4, 13, 258)-Net over F₁₂₈ — Constructive and digital

Digital (4, 13, 258)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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]
2. digital (0, 9, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(4, 13, 259)-Net in Base 128 — Constructive

(4, 13, 259)-net in base 128, using

3 times m-reduction [i] based on (4, 16, 259)-net in base 128, using
- base change [i] based on digital (2, 14, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(4, 13, 321)-Net in Base 128

(4, 13, 321)-net in base 128, using

3 times m-reduction [i] based on (4, 16, 321)-net in base 128, using
- base change [i] based on digital (2, 14, 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]

(4, 13, 36547)-Net in Base 128 — Upper bound on s

There is no (4, 13, 36548)-net in base 128, because

1 times m-reduction [i] would yield (4, 12, 36548)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 19 343314 759937 399130 481584 > 128¹² [i]