MinT - Best Known (67−9, 67, s)-Nets in Base 8

Best Known (67−9, 67, s)-Nets in Base 8

(67−9, 67, 4194300)-Net over F₈ — Constructive and digital

Digital (58, 67, 4194300)-net over F₈, using

8¹ times duplication [i] based on digital (57, 66, 4194300)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁶⁶, 4194300, F₈, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁶⁶, 8388601, F₈, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8⁶⁶, 8388602, F₈, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
      - trace code [i] based on linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(67−9, 67, large)-Net over F₈ — Digital

Digital (58, 67, large)-net over F₈, using

8² times duplication [i] based on digital (56, 65, large)-net over F₈, using
- t-expansion [i] based on digital (55, 65, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁵, large, F₈, 10) (dual of [large, large−65, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(67−9, 67, large)-Net in Base 8 — Upper bound on s

There is no (58, 67, large)-net in base 8, because

7 times m-reduction [i] would yield (58, 60, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]