MinT - Best Known (67, 67+22, s)-Nets in Base 3

Best Known (67, 67+22, s)-Nets in Base 3

(67, 67+22, 328)-Net over F₃ — Constructive and digital

Digital (67, 89, 328)-net over F₃, using

3¹ times duplication [i] based on digital (66, 88, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 22, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(67, 67+22, 505)-Net over F₃ — Digital

Digital (67, 89, 505)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸⁹, 505, F₃, 22) (dual of [505, 416, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸⁹, 745, F₃, 22) (dual of [745, 656, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [i] based on
    1. linear OA(3⁸⁵, 729, F₃, 22) (dual of [729, 644, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(3⁷³, 729, F₃, 19) (dual of [729, 656, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3⁴, 16, F₃, 2) (dual of [16, 12, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(67, 67+22, 17785)-Net in Base 3 — Upper bound on s

There is no (67, 89, 17786)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 909739 603206 494282 279839 821517 439420 868001 > 3⁸⁹ [i]