MinT - Best Known (124, 124+117, s)-Nets in Base 3

Best Known (124, 124+117, s)-Nets in Base 3

(124, 124+117, 80)-Net over F₃ — Constructive and digital

Digital (124, 241, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 79, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 162, 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]

(124, 124+117, 128)-Net over F₃ — Digital

Digital (124, 241, 128)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+117, 1001)-Net in Base 3 — Upper bound on s

There is no (124, 241, 1002)-net in base 3, because

1 times m-reduction [i] would yield (124, 240, 1002)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 268420 364078 690548 516314 770625 073454 064726 885471 307469 980370 135488 286560 005936 490921 759845 483428 976550 822807 956597 > 3²⁴⁰ [i]