MinT - Best Known (85, 89, s)-Nets in Base 3

Best Known (85, 89, s)-Nets in Base 3

(85, 89, large)-Net over F₃ — Constructive and digital

Digital (85, 89, large)-net over F₃, using

3² times duplication [i] based on digital (83, 87, large)-net over F₃, using
- t-expansion [i] based on digital (81, 87, large)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁸⁷, large, F₃, 6, 6), using
    - appending kth column [i] based on linear OOA(3⁸⁷, large, F₃, 5, 6), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3³, 4, F₃, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(5;17,3) [i]
        linear OOA(3⁸⁴, 8388602, F₃, 5, 6) (dual of [(8388602, 5), 41942926, 7]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(3⁸⁴, large, F₃, 3, 6), using
        trace code [i] based on linear OOA(81²¹, 2796201, F₈₁, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81²¹, large, F₈₁, 6) (dual of [large, large−21, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(85, 89, large)-Net in Base 3 — Upper bound on s

There is no (85, 89, large)-net in base 3, because

2 times m-reduction [i] would yield (85, 87, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]