MinT - Best Known (47−11, 47, s)-Nets in Base 3

Best Known (47−11, 47, s)-Nets in Base 3

(47−11, 47, 328)-Net over F₃ — Constructive and digital

Digital (36, 47, 328)-net over F₃, using

1 times m-reduction [i] based on digital (36, 48, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 12, 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]

(47−11, 47, 562)-Net over F₃ — Digital

Digital (36, 47, 562)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴⁷, 562, F₃, 11) (dual of [562, 515, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁴⁷, 745, F₃, 11) (dual of [745, 698, 12]-code), using
  - constructionÂ X applied to C_e(10) âŠ‚ C_e(7) [i] based on
    1. linear OA(3⁴³, 729, F₃, 11) (dual of [729, 686, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(3³¹, 729, F₃, 8) (dual of [729, 698, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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]

(47−11, 47, 31934)-Net in Base 3 — Upper bound on s

There is no (36, 47, 31935)-net in base 3, because

1 times m-reduction [i] would yield (36, 46, 31935)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8863 582266 822711 759735 > 3⁴⁶ [i]