MinT - Best Known (128−27, 128, s)-Nets in Base 3

Best Known (128−27, 128, s)-Nets in Base 3

(128−27, 128, 640)-Net over F₃ — Constructive and digital

Digital (101, 128, 640)-net over F₃, using

trace code for nets [i] based on digital (5, 32, 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]

(128−27, 128, 1327)-Net over F₃ — Digital

Digital (101, 128, 1327)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁸, 1327, F₃, 27) (dual of [1327, 1199, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁸, 2203, F₃, 27) (dual of [2203, 2075, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [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(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(128−27, 128, 129850)-Net in Base 3 — Upper bound on s

There is no (101, 128, 129851)-net in base 3, because

1 times m-reduction [i] would yield (101, 127, 129851)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 930447 061212 294060 772890 433674 416452 274716 208775 658657 001823 > 3¹²⁷ [i]