MinT - Best Known (22−17, 22, s)-Nets in Base 128

Best Known (22−17, 22, s)-Nets in Base 128

(22−17, 22, 216)-Net over F₁₂₈ — Constructive and digital

Digital (5, 22, 216)-net over F₁₂₈, using

net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
  - a function field by SÃ©mirat [i]

(22−17, 22, 227)-Net over F₁₂₈ — Digital

Digital (5, 22, 227)-net over F₁₂₈, using

net from sequence [i] based on digital (5, 226)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 227, using
  - a curve from the manYPoints database [i]

(22−17, 22, 259)-Net in Base 128 — Constructive

(5, 22, 259)-net in base 128, using

2 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
- base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(22−17, 22, 321)-Net in Base 128

(5, 22, 321)-net in base 128, using

2 times m-reduction [i] based on (5, 24, 321)-net in base 128, using
- base change [i] based on digital (2, 21, 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]

(22−17, 22, 10073)-Net in Base 128 — Upper bound on s

There is no (5, 22, 10074)-net in base 128, because

1 times m-reduction [i] would yield (5, 21, 10074)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 178 547312 981995 837543 775002 024185 994329 074335 > 128²¹ [i]