MinT - Best Known (244−119, 244, s)-Nets in Base 3

Best Known (244−119, 244, s)-Nets in Base 3

(244−119, 244, 80)-Net over F₃ — Constructive and digital

Digital (125, 244, 80)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 80, 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, 164, 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]

(244−119, 244, 128)-Net over F₃ — Digital

Digital (125, 244, 128)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−119, 244, 995)-Net in Base 3 — Upper bound on s

There is no (125, 244, 996)-net in base 3, because

1 times m-reduction [i] would yield (125, 243, 996)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88 416785 950445 877195 113589 916526 423648 837653 592615 215899 828788 972397 090136 349147 921671 027460 116628 828567 940268 440561 > 3²⁴³ [i]