MinT - Best Known (220−99, 220, s)-Nets in Base 3

Best Known (220−99, 220, s)-Nets in Base 3

(220−99, 220, 85)-Net over F₃ — Constructive and digital

Digital (121, 220, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 76, 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]
2. digital (45, 144, 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]

(220−99, 220, 146)-Net over F₃ — Digital

Digital (121, 220, 146)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−99, 220, 1248)-Net in Base 3 — Upper bound on s

There is no (121, 220, 1249)-net in base 3, because

1 times m-reduction [i] would yield (121, 219, 1249)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 313 662241 630120 795572 003266 082456 397876 096985 148473 819205 798129 788305 585909 715644 809390 164624 931060 115203 > 3²¹⁹ [i]