MinT - Best Known (47, 54, s)-Nets in Base 8

Best Known (47, 54, s)-Nets in Base 8

(47, 54, 5592466)-Net over F₈ — Constructive and digital

Digital (47, 54, 5592466)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁵⁴, 5592466, F₈, 9, 7) (dual of [(5592466, 9), 50332140, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8⁵⁴, 5592467, F₈, 3, 7) (dual of [(5592467, 3), 16777347, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
    2. linear OOA(8⁵⁰, 5592402, F₈, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
      - trace code [i] based on linear OOA(64²⁵, 2796201, F₆₄, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(47, 54, large)-Net over F₈ — Digital

Digital (47, 54, large)-net over F₈, using

8⁵ times duplication [i] based on digital (42, 49, large)-net over F₈, using
- Gilbertâ€“VarÅ¡amov bound [i]

(47, 54, large)-Net in Base 8 — Upper bound on s

There is no (47, 54, large)-net in base 8, because

5 times m-reduction [i] would yield (47, 49, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]