MinT - Best Known (160, 181, s)-Nets in Base 3

Best Known (160, 181, s)-Nets in Base 3

(160, 181, 53153)-Net over F₃ — Constructive and digital

Digital (160, 181, 53153)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (147, 168, 53143)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁸, 53143, F₃, 21, 21) (dual of [(53143, 21), 1115835, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁶⁸, 531431, F₃, 21) (dual of [531431, 531263, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁸, 531440, F₃, 21) (dual of [531440, 531272, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁶⁹, 531441, F₃, 22) (dual of [531441, 531272, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(160, 181, 209783)-Net over F₃ — Digital

Digital (160, 181, 209783)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸¹, 209783, F₃, 2, 21) (dual of [(209783, 2), 419385, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁸¹, 265751, F₃, 2, 21) (dual of [(265751, 2), 531321, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁸¹, 531502, F₃, 21) (dual of [531502, 531321, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(3¹⁸⁰, 531501, F₃, 21) (dual of [531501, 531321, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
        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]
        linear OA(3¹²¹, 531442, F₃, 15) (dual of [531442, 531321, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        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]

(160, 181, large)-Net in Base 3 — Upper bound on s

There is no (160, 181, large)-net in base 3, because

19 times m-reduction [i] would yield (160, 162, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]