MinT - Best Known (55−7, 55, s)-Nets in Base 9

Best Known (55−7, 55, s)-Nets in Base 9

(55−7, 55, 5592613)-Net over F₉ — Constructive and digital

Digital (48, 55, 5592613)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁵⁵, 5592613, F₉, 9, 7) (dual of [(5592613, 9), 50333462, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9⁵⁵, 5592614, F₉, 3, 7) (dual of [(5592614, 3), 16777787, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(9⁵, 212, F₉, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(9⁵, 212, F₉, 2, 3) (dual of [(212, 2), 419, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁵, 212, F₉, 3) (dual of [212, 207, 4]-code or 212-cap in PG(4,9)), using
        caps completed using a computer search [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]

(55−7, 55, large)-Net over F₉ — Digital

Digital (48, 55, large)-net over F₉, using

9¹ 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]

(55−7, 55, large)-Net in Base 9 — Upper bound on s

There is no (48, 55, large)-net in base 9, because

5 times m-reduction [i] would yield (48, 50, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]