MinT - Best Known (230−27, 230, s)-Nets in Base 3

Best Known (230−27, 230, s)-Nets in Base 3

(230−27, 230, 40886)-Net over F₃ — Constructive and digital

Digital (203, 230, 40886)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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]
2. digital (189, 216, 40879)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁶, 40879, F₃, 27, 27) (dual of [(40879, 27), 1103517, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(3²¹⁶, 531428, F₃, 27) (dual of [531428, 531212, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁶, 531440, F₃, 27) (dual of [531440, 531224, 28]-code), using
        1 times truncation [i] based on linear OA(3²¹⁷, 531441, F₃, 28) (dual of [531441, 531224, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(230−27, 230, 177167)-Net over F₃ — Digital

Digital (203, 230, 177167)-net over F₃, using

3² times duplication [i] based on digital (201, 228, 177167)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁸, 177167, F₃, 3, 27) (dual of [(177167, 3), 531273, 28]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²²⁸, 531501, F₃, 27) (dual of [531501, 531273, 28]-code), using
    - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(3²¹⁷, 531442, F₃, 27) (dual of [531442, 531225, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      2. linear OA(3¹⁶⁹, 531442, F₃, 21) (dual of [531442, 531273, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      3. linear OA(3¹¹, 59, F₃, 5) (dual of [59, 48, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(230−27, 230, large)-Net in Base 3 — Upper bound on s

There is no (203, 230, large)-net in base 3, because

25 times m-reduction [i] would yield (203, 205, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]