MinT - Best Known (212−27, 212, s)-Nets in Base 3

Best Known (212−27, 212, s)-Nets in Base 3

(212−27, 212, 13633)-Net over F₃ — Constructive and digital

Digital (185, 212, 13633)-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 (171, 198, 13626)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁸, 13626, F₃, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(3¹⁹⁸, 177139, F₃, 27) (dual of [177139, 176941, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁸, 177146, F₃, 27) (dual of [177146, 176948, 28]-code), using
        1 times truncation [i] based on linear OA(3¹⁹⁹, 177147, F₃, 28) (dual of [177147, 176948, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(212−27, 212, 73284)-Net over F₃ — Digital

Digital (185, 212, 73284)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹², 73284, F₃, 2, 27) (dual of [(73284, 2), 146356, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹², 88602, F₃, 2, 27) (dual of [(88602, 2), 176992, 28]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3²¹¹, 88602, F₃, 2, 27) (dual of [(88602, 2), 176993, 28]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹¹, 177204, F₃, 27) (dual of [177204, 176993, 28]-code), using
      - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
        linear OA(3¹⁹⁹, 177148, F₃, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(3¹⁵⁵, 177148, F₃, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(212−27, 212, large)-Net in Base 3 — Upper bound on s

There is no (185, 212, large)-net in base 3, because

25 times m-reduction [i] would yield (185, 187, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (212−27, 212, s)-Nets in Base 3

(212−27, 212, 13633)-Net over F3 — Constructive and digital

(212−27, 212, 73284)-Net over F3 — Digital

(212−27, 212, large)-Net in Base 3 — Upper bound on s

(212−27, 212, 13633)-Net over F₃ — Constructive and digital

(212−27, 212, 73284)-Net over F₃ — Digital