MinT - Best Known (50−5, 50, s)-Nets in Base 8

Best Known (50−5, 50, s)-Nets in Base 8

(50−5, 50, large)-Net over F₈ — Constructive and digital

Digital (45, 50, large)-net over F₈, using

8¹² times duplication [i] based on digital (33, 38, large)-net over F₈, using
- trace code for nets [i] based on digital (14, 19, 4194365)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64¹⁹, 4194365, F₆₄, 6, 5) (dual of [(4194365, 6), 25166171, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(64¹⁹, 4194366, F₆₄, 2, 5) (dual of [(4194366, 2), 8388713, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(64², 65, F₆₄, 2, 2) (dual of [(65, 2), 128, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;128,64) [i]
        linear OOA(64¹⁷, 4194301, F₆₄, 2, 5) (dual of [(4194301, 2), 8388585, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64¹⁷, 8388602, F₆₄, 5) (dual of [8388602, 8388585, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(50−5, 50, large)-Net in Base 8 — Upper bound on s

There is no (45, 50, large)-net in base 8, because

3 times m-reduction [i] would yield (45, 47, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]