MinT - Best Known (8, 8+48, s)-Nets in Base 128

Best Known (8, 8+48, s)-Nets in Base 128

(8, 8+48, 216)-Net over F₁₂₈ — Constructive and digital

Digital (8, 56, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 56, 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+48, 258)-Net in Base 128 — Constructive

(8, 56, 258)-net in base 128, using

base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(8, 8+48, 276)-Net over F₁₂₈ — Digital

Digital (8, 56, 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+48, 289)-Net in Base 128

(8, 56, 289)-net in base 128, using

base change [i] based on digital (1, 49, 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]

(8, 8+48, 6361)-Net in Base 128 — Upper bound on s

There is no (8, 56, 6362)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 10091 546614 767338 320769 041290 254725 437607 297206 306280 523861 580776 957951 213887 404545 038564 411001 879016 270452 915904 714773 > 128⁵⁶ [i]