MinT - Best Known (12, 12+20, s)-Nets in Base 128

Best Known (12, 12+20, s)-Nets in Base 128

(12, 12+20, 300)-Net over F₁₂₈ — Constructive and digital

Digital (12, 32, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (1, 21, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(12, 12+20, 321)-Net over F₁₂₈ — Digital

Digital (12, 32, 321)-net over F₁₂₈, using

net from sequence [i] based on digital (12, 320)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 12 and N(F) ≥ 321, using
  - a curve from the manYPoints database [i]

(12, 12+20, 513)-Net in Base 128

(12, 32, 513)-net in base 128, using

base change [i] based on digital (8, 28, 513)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(12, 12+20, 197348)-Net in Base 128 — Upper bound on s

There is no (12, 32, 197349)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 26 960270 025539 004501 384048 052721 047995 363006 549739 859694 578115 148376 > 128³² [i]