MinT - Best Known (129−5, 129, s)-Nets in Base 3

Best Known (129−5, 129, s)-Nets in Base 3

(129−5, 129, large)-Net over F₃ — Constructive and digital

Digital (124, 129, large)-net over F₃, using

3 times m-reduction [i] based on digital (124, 132, large)-net over F₃, using
- trace code for nets [i] based on digital (25, 33, 2097231)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³³, 2097231, F₈₁, 12, 8) (dual of [(2097231, 12), 25166739, 9]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(81³³, 2097232, F₈₁, 4, 8) (dual of [(2097232, 4), 8388895, 9]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(81⁴, 82, F₈₁, 4, 4) (dual of [(82, 4), 324, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;324,81) [i]
        linear OOA(81²⁹, 2097150, F₈₁, 4, 8) (dual of [(2097150, 4), 8388571, 9]-NRT-code), using
        OOA 4-folding [i] based on linear OA(81²⁹, 8388600, F₈₁, 8) (dual of [8388600, 8388571, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(81²⁹, large, F₈₁, 8) (dual of [large, large−29, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(129−5, 129, large)-Net in Base 3 — Upper bound on s

There is no (124, 129, large)-net in base 3, because

3 times m-reduction [i] would yield (124, 126, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]