MinT - Best Known (8, 8+40, s)-Nets in Base 128

Best Known (8, 8+40, s)-Nets in Base 128

(8, 8+40, 216)-Net over F₁₂₈ — Constructive and digital

Digital (8, 48, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 48, 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]

(8, 8+40, 259)-Net in Base 128 — Constructive

(8, 48, 259)-net in base 128, using

base change [i] based on digital (2, 42, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(8, 8+40, 276)-Net over F₁₂₈ — Digital

Digital (8, 48, 276)-net over F₁₂₈, using

net from sequence [i] based on digital (8, 275)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 8 and N(F) ≥ 276, using
  - a curve from the manYPoints database [i]

(8, 8+40, 321)-Net in Base 128

(8, 48, 321)-net in base 128, using

base change [i] based on digital (2, 42, 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]

(8, 8+40, 7451)-Net in Base 128 — Upper bound on s

There is no (8, 48, 7452)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 140169 382474 648716 389907 050991 371451 653482 546167 318212 750573 258486 974450 730629 087978 412726 246855 293246 > 128⁴⁸ [i]