MinT - Best Known (128−25, 128, s)-Nets in Base 3

Best Known (128−25, 128, s)-Nets in Base 3

(128−25, 128, 688)-Net over F₃ — Constructive and digital

Digital (103, 128, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 32, 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]

(128−25, 128, 2011)-Net over F₃ — Digital

Digital (103, 128, 2011)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁸, 2011, F₃, 25) (dual of [2011, 1883, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁸, 2231, F₃, 25) (dual of [2231, 2103, 26]-code), using
  - 5 times code embedding in larger space [i] based on linear OA(3¹²³, 2226, F₃, 25) (dual of [2226, 2103, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. 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]
      2. linear OA(3⁸⁵, 2188, F₃, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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]

(128−25, 128, 296379)-Net in Base 3 — Upper bound on s

There is no (103, 128, 296380)-net in base 3, because

1 times m-reduction [i] would yield (103, 127, 296380)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 930175 251716 372374 784045 105133 820491 588212 141951 598817 740609 > 3¹²⁷ [i]