MinT - Best Known (22, 22+10, s)-Nets in Base 3

Best Known (22, 22+10, s)-Nets in Base 3

(22, 22+10, 84)-Net over F₃ — Constructive and digital

Digital (22, 32, 84)-net over F₃, using

1 times m-reduction [i] based on digital (22, 33, 84)-net over F₃, using
- trace code for nets [i] based on digital (0, 11, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]

(22, 22+10, 127)-Net over F₃ — Digital

Digital (22, 32, 127)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³², 127, F₃, 10) (dual of [127, 95, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3³², 249, F₃, 10) (dual of [249, 217, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
    1. linear OA(3³¹, 243, F₃, 10) (dual of [243, 212, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(3²⁶, 243, F₃, 8) (dual of [243, 217, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3¹, 6, F₃, 1) (dual of [6, 5, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(22, 22+10, 1469)-Net in Base 3 — Upper bound on s

There is no (22, 32, 1470)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1858 592315 849109 > 3³² [i]