MinT - Best Known (50, 57, s)-Nets in Base 9

Best Known (50, 57, s)-Nets in Base 9

(50, 57, 5599124)-Net over F₉ — Constructive and digital

Digital (50, 57, 5599124)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁵⁷, 5599124, F₉, 9, 7) (dual of [(5599124, 9), 50392059, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9⁵⁷, 5599125, F₉, 3, 7) (dual of [(5599125, 3), 16797318, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(9⁷, 6723, F₉, 3, 3) (dual of [(6723, 3), 20162, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(9⁷, 6723, F₉, 2, 3) (dual of [(6723, 2), 13439, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁷, 6723, F₉, 3) (dual of [6723, 6716, 4]-code or 6723-cap in PG(6,9)), using
        product of two caps with tangent hyperplane [i]
    2. linear OOA(9⁵⁰, 5592402, F₉, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
      - trace code [i] based on linear OOA(81²⁵, 2796201, F₈₁, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81²⁵, large, F₈₁, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(50, 57, large)-Net over F₉ — Digital

Digital (50, 57, large)-net over F₉, using

9³ times duplication [i] based on digital (47, 54, large)-net over F₉, using
- t-expansion [i] based on digital (46, 54, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(50, 57, large)-Net in Base 9 — Upper bound on s

There is no (50, 57, large)-net in base 9, because

5 times m-reduction [i] would yield (50, 52, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]