MinT - Best Known (124, 124+31, s)-Nets in Base 3

Best Known (124, 124+31, s)-Nets in Base 3

(124, 124+31, 688)-Net over F₃ — Constructive and digital

Digital (124, 155, 688)-net over F₃, using

1 times m-reduction [i] based on digital (124, 156, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 39, 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]

(124, 124+31, 1967)-Net over F₃ — Digital

Digital (124, 155, 1967)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁵, 1967, F₃, 31) (dual of [1967, 1812, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁵, 2230, F₃, 31) (dual of [2230, 2075, 32]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3¹⁵¹, 2226, F₃, 31) (dual of [2226, 2075, 32]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (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(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]

(124, 124+31, 254189)-Net in Base 3 — Upper bound on s

There is no (124, 155, 254190)-net in base 3, because

1 times m-reduction [i] would yield (124, 154, 254190)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 970834 072819 127686 246466 519832 072303 870065 754549 668163 805981 053387 574441 > 3¹⁵⁴ [i]