MinT - Best Known (100, 125, s)-Nets in Base 3

Best Known (100, 125, s)-Nets in Base 3

(100, 125, 640)-Net over F₃ — Constructive and digital

Digital (100, 125, 640)-net over F₃, using

3¹ times duplication [i] based on digital (99, 124, 640)-net over F₃, using
- t-expansion [i] based on digital (98, 124, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 31, 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]

(100, 125, 1740)-Net over F₃ — Digital

Digital (100, 125, 1740)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁵, 1740, F₃, 25) (dual of [1740, 1615, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁵, 2201, F₃, 25) (dual of [2201, 2076, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([1,12]) [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(3¹¹², 2188, F₃, 12) (dual of [2188, 2076, 13]-code), using the narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(3¹², 13, F₃, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
      - dual of repetition code with length 13 [i]

(100, 125, 225196)-Net in Base 3 — Upper bound on s

There is no (100, 125, 225197)-net in base 3, because

1 times m-reduction [i] would yield (100, 124, 225197)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 145559 419170 857449 817148 341602 353583 109612 804086 287705 190001 > 3¹²⁴ [i]