MinT - Best Known (189−24, 189, s)-Nets in Base 3

Best Known (189−24, 189, s)-Nets in Base 3

(189−24, 189, 14769)-Net over F₃ — Constructive and digital

Digital (165, 189, 14769)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 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 (152, 176, 14762)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁷⁶, 14762, F₃, 24, 24) (dual of [(14762, 24), 354112, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(3¹⁷⁶, 177144, F₃, 24) (dual of [177144, 176968, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁷⁶, 177147, F₃, 24) (dual of [177147, 176971, 25]-code), using
        1 times truncation [i] based on linear OA(3¹⁷⁷, 177148, F₃, 25) (dual of [177148, 176971, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(189−24, 189, 76913)-Net over F₃ — Digital

Digital (165, 189, 76913)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁹, 76913, F₃, 2, 24) (dual of [(76913, 2), 153637, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁸⁹, 88602, F₃, 2, 24) (dual of [(88602, 2), 177015, 25]-NRT-code), using
  - strength reduction [i] based on linear OOA(3¹⁸⁹, 88602, F₃, 2, 25) (dual of [(88602, 2), 177015, 26]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁸⁹, 177204, F₃, 25) (dual of [177204, 177015, 26]-code), using
      - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
        linear OA(3¹⁷⁷, 177148, F₃, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(3¹³³, 177148, F₃, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3¹², 56, F₃, 5) (dual of [56, 44, 6]-code), using
        (u,Â u+v)-construction [i] based on
        linear OA(3⁴, 28, F₃, 2) (dual of [28, 24, 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]
        linear OA(3⁸, 28, F₃, 5) (dual of [28, 20, 6]-code), using
        dual code (with bound on d by constructionÂ Y1) [i] based on
        linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code), using
        contraction [i] based on linear OA(3⁴⁸, 56, F₃, 29) (dual of [56, 8, 30]-code), using the BCH-code C(I) with length 56Â | 3⁶âˆ’1, defining interval IÂ =Â {0,1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
        nonexistence of linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), because
        2 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(189−24, 189, large)-Net in Base 3 — Upper bound on s

There is no (165, 189, large)-net in base 3, because

22 times m-reduction [i] would yield (165, 167, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (189−24, 189, s)-Nets in Base 3

(189−24, 189, 14769)-Net over F3 — Constructive and digital

(189−24, 189, 76913)-Net over F3 — Digital

(189−24, 189, large)-Net in Base 3 — Upper bound on s

(189−24, 189, 14769)-Net over F₃ — Constructive and digital

(189−24, 189, 76913)-Net over F₃ — Digital