MinT - Best Known (139−27, 139, s)-Nets in Base 3

Best Known (139−27, 139, s)-Nets in Base 3

(139−27, 139, 688)-Net over F₃ — Constructive and digital

Digital (112, 139, 688)-net over F₃, using

1 times m-reduction [i] based on digital (112, 140, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 35, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(139−27, 139, 2166)-Net over F₃ — Digital

Digital (112, 139, 2166)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁹, 2166, F₃, 27) (dual of [2166, 2027, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁹, 2228, F₃, 27) (dual of [2228, 2089, 28]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3¹³⁷, 2226, F₃, 27) (dual of [2226, 2089, 28]-code), using
    - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(3¹²⁷, 2188, F₃, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      2. 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]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(139−27, 139, 328991)-Net in Base 3 — Upper bound on s

There is no (112, 139, 328992)-net in base 3, because

1 times m-reduction [i] would yield (112, 138, 328992)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 696199 198910 314007 826707 242904 206800 029375 603033 102373 082380 705857 > 3¹³⁸ [i]