MinT - Best Known (157−77, 157, s)-Nets in Base 3

Best Known (157−77, 157, s)-Nets in Base 3

(157−77, 157, 65)-Net over F₃ — Constructive and digital

Digital (80, 157, 65)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 53, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 104, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(157−77, 157, 86)-Net over F₃ — Digital

Digital (80, 157, 86)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−77, 157, 646)-Net in Base 3 — Upper bound on s

There is no (80, 157, 647)-net in base 3, because

1 times m-reduction [i] would yield (80, 156, 647)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 278 422229 096787 792163 426572 553029 623336 334002 285463 904624 647991 807506 675333 > 3¹⁵⁶ [i]