MinT - Best Known (231−103, 231, s)-Nets in Base 3

Best Known (231−103, 231, s)-Nets in Base 3

(231−103, 231, 86)-Net over F₃ — Constructive and digital

Digital (128, 231, 86)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (32, 83, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 148, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(231−103, 231, 156)-Net over F₃ — Digital

Digital (128, 231, 156)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−103, 231, 1358)-Net in Base 3 — Upper bound on s

There is no (128, 231, 1359)-net in base 3, because

1 times m-reduction [i] would yield (128, 230, 1359)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 56 567198 702630 359946 621774 142293 618481 027861 939833 109309 966004 151284 631923 660491 956029 199553 667492 428215 097819 > 3²³⁰ [i]