MinT - Best Known (227−35, 227, s)-Nets in Base 3

Best Known (227−35, 227, s)-Nets in Base 3

(227−35, 227, 1494)-Net over F₃ — Constructive and digital

Digital (192, 227, 1494)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 14)-net over F₃, using
  - net from sequence [i] based on digital (6, 13)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 6 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (169, 204, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 51, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(227−35, 227, 12155)-Net over F₃ — Digital

Digital (192, 227, 12155)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁷, 12155, F₃, 35) (dual of [12155, 11928, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁷, 19747, F₃, 35) (dual of [19747, 19520, 36]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3²²³, 19743, F₃, 35) (dual of [19743, 19520, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(27) [i] based on
      1. linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      2. linear OA(3¹⁶³, 19683, F₃, 28) (dual of [19683, 19520, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3¹⁵, 60, F₃, 6) (dual of [60, 45, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(227−35, 227, 7903517)-Net in Base 3 — Upper bound on s

There is no (192, 227, 7903518)-net in base 3, because

1 times m-reduction [i] would yield (192, 226, 7903518)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 675156 446450 649810 344892 456530 935036 864097 609772 093288 586151 901006 764027 924714 771696 428923 027412 679303 044125 > 3²²⁶ [i]