MinT - Best Known (93, 93+81, s)-Nets in Base 3

Best Known (93, 93+81, s)-Nets in Base 3

(93, 93+81, 73)-Net over F₃ — Constructive and digital

Digital (93, 174, 73)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 66, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 108, 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]

(93, 93+81, 108)-Net over F₃ — Digital

Digital (93, 174, 108)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(93, 93+81, 873)-Net in Base 3 — Upper bound on s

There is no (93, 174, 874)-net in base 3, because

1 times m-reduction [i] would yield (93, 173, 874)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35046 186745 257857 882465 597141 449666 681839 979094 466726 457442 026094 665432 288282 868241 > 3¹⁷³ [i]