MinT - Best Known (139, 139+5, s)-Nets in Base 3

Best Known (139, 139+5, s)-Nets in Base 3

(139, 139+5, large)-Net over F₃ — Constructive and digital

Digital (139, 144, large)-net over F₃, using

4 times m-reduction [i] based on digital (139, 148, large)-net over F₃, using
- trace code for nets [i] based on digital (28, 37, 2097231)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³⁷, 2097231, F₈₁, 12, 9) (dual of [(2097231, 12), 25166735, 10]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(81³⁷, 2097232, F₈₁, 4, 9) (dual of [(2097232, 4), 8388891, 10]-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(81³³, 2097150, F₈₁, 4, 9) (dual of [(2097150, 4), 8388567, 10]-NRT-code), using
        OOA 4-folding [i] based on linear OA(81³³, 8388600, F₈₁, 9) (dual of [8388600, 8388567, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(81³³, large, F₈₁, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(139, 139+5, large)-Net in Base 3 — Upper bound on s

There is no (139, 144, large)-net in base 3, because

3 times m-reduction [i] would yield (139, 141, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]