MinT - Best Known (48, 48+46, s)-Nets in Base 3

Best Known (48, 48+46, s)-Nets in Base 3

(48, 48+46, 48)-Net over F₃ — Constructive and digital

Digital (48, 94, 48)-net over F₃, using

t-expansion [i] based on digital (45, 94, 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]

(48, 48+46, 58)-Net over F₃ — Digital

Digital (48, 94, 58)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹⁴, 58, F₃, 5, 46) (dual of [(58, 5), 196, 47]-NRT-code), using
- strength reduction [i] based on linear OOA(3⁹⁴, 58, F₃, 5, 47) (dual of [(58, 5), 196, 48]-NRT-code), using
  - constructionÂ X applied to AG(5;F,227P) âŠ‚ AG(5;F,235P) [i] based on
    1. linear OOA(3⁸⁷, 55, F₃, 5, 47) (dual of [(55, 5), 188, 48]-NRT-code), using algebraic-geometric NRT-code AG(5;F,227P) [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
      - a curve from the manYPoints database [i]
    2. linear OOA(3⁷⁹, 55, F₃, 5, 39) (dual of [(55, 5), 196, 40]-NRT-code), using algebraic-geometric NRT-code AG(5;F,235P) [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56 (see above)
    3. linear OOA(3⁷, 3, F₃, 5, 7) (dual of [(3, 5), 8, 8]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(5;8,3) [i]

(48, 48+46, 398)-Net in Base 3 — Upper bound on s

There is no (48, 94, 399)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 741 228505 962149 109879 777006 399918 680789 255955 > 3⁹⁴ [i]