MinT - Best Known (67, 67+12, s)-Nets in Base 3

Best Known (67, 67+12, s)-Nets in Base 3

(67, 67+12, 3287)-Net over F₃ — Constructive and digital

Digital (67, 79, 3287)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - a shift-net [i]
2. digital (60, 72, 3280)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁷², 3280, F₃, 12, 12) (dual of [(3280, 12), 39288, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(3⁷², 19680, F₃, 12) (dual of [19680, 19608, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷², 19683, F₃, 12) (dual of [19683, 19611, 13]-code), using
        1 times truncation [i] based on linear OA(3⁷³, 19684, F₃, 13) (dual of [19684, 19611, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(67, 67+12, 11917)-Net over F₃ — Digital

Digital (67, 79, 11917)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁹, 11917, F₃, 12) (dual of [11917, 11838, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁹, 19716, F₃, 12) (dual of [19716, 19637, 13]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(7) [i] based on
    1. linear OA(3⁷³, 19683, F₃, 13) (dual of [19683, 19610, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(3⁴⁶, 19683, F₃, 8) (dual of [19683, 19637, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3⁶, 33, F₃, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(67, 67+12, 2866076)-Net in Base 3 — Upper bound on s

There is no (67, 79, 2866077)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49 269664 143285 225298 544349 444834 871345 > 3⁷⁹ [i]