MinT - Best Known (86, 86+21, s)-Nets in Base 3

Best Known (86, 86+21, s)-Nets in Base 3

(86, 86+21, 640)-Net over F₃ — Constructive and digital

Digital (86, 107, 640)-net over F₃, using

1 times m-reduction [i] based on digital (86, 108, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 27, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(86, 86+21, 1802)-Net over F₃ — Digital

Digital (86, 107, 1802)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁷, 1802, F₃, 21) (dual of [1802, 1695, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁷, 2216, F₃, 21) (dual of [2216, 2109, 22]-code), using
  - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(3⁹⁹, 2188, F₃, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    2. linear OA(3⁷¹, 2188, F₃, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    3. 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]

(86, 86+21, 258473)-Net in Base 3 — Upper bound on s

There is no (86, 107, 258474)-net in base 3, because

1 times m-reduction [i] would yield (86, 106, 258474)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 375 720747 336596 403728 734437 134574 119319 200995 726101 > 3¹⁰⁶ [i]

Best Known (86, 86+21, s)-Nets in Base 3

(86, 86+21, 640)-Net over F3 — Constructive and digital

(86, 86+21, 1802)-Net over F3 — Digital

(86, 86+21, 258473)-Net in Base 3 — Upper bound on s

(86, 86+21, 640)-Net over F₃ — Constructive and digital

(86, 86+21, 1802)-Net over F₃ — Digital