MinT - Best Known (56, 109, s)-Nets in Base 3

Best Known (56, 109, s)-Nets in Base 3

(56, 109, 56)-Net over F₃ — Constructive and digital

Digital (56, 109, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 41, 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 (15, 68, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(56, 109, 66)-Net over F₃ — Digital

Digital (56, 109, 66)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁰⁹, 66, F₃, 5, 53) (dual of [(66, 5), 221, 54]-NRT-code), using
- constructionÂ X applied to AG(5;F,261P) âŠ‚ AG(5;F,269P) [i] based on
  1. linear OOA(3¹⁰², 63, F₃, 5, 53) (dual of [(63, 5), 213, 54]-NRT-code), using algebraic-geometric NRT-code AG(5;F,261P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]
  2. linear OOA(3⁹⁴, 63, F₃, 5, 45) (dual of [(63, 5), 221, 46]-NRT-code), using algebraic-geometric NRT-code AG(5;F,269P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64 (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]

(56, 109, 481)-Net in Base 3 — Upper bound on s

There is no (56, 109, 482)-net in base 3, because

1 times m-reduction [i] would yield (56, 108, 482)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3555 352054 719713 831358 038290 382006 008550 859312 765765 > 3¹⁰⁸ [i]