MinT - Best Known (243−111, 243, s)-Nets in Base 3

Best Known (243−111, 243, s)-Nets in Base 3

(243−111, 243, 86)-Net over F₃ — Constructive and digital

Digital (132, 243, 86)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (32, 87, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 156, 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]

(243−111, 243, 153)-Net over F₃ — Digital

Digital (132, 243, 153)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−111, 243, 1287)-Net in Base 3 — Upper bound on s

There is no (132, 243, 1288)-net in base 3, because

1 times m-reduction [i] would yield (132, 242, 1288)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 660752 895441 234802 127925 779901 103211 118123 645583 134099 393973 947854 973131 019333 754631 806293 227249 950898 260535 827297 > 3²⁴² [i]