MinT - Best Known (68, 68+7, s)-Nets in Base 8

Best Known (68, 68+7, s)-Nets in Base 8

(68, 68+7, large)-Net over F₈ — Constructive and digital

Digital (68, 75, large)-net over F₈, using

8¹¹ times duplication [i] based on digital (57, 64, large)-net over F₈, using
- trace code for nets [i] based on digital (25, 32, 5592401)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³², 5592401, F₆₄, 9, 7) (dual of [(5592401, 9), 50331577, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(64³², 5592402, F₆₄, 3, 7) (dual of [(5592402, 3), 16777174, 8]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(64⁷, large, F₆₄, 3, 3), using
        appending kth column [i] based on linear OOA(64⁷, large, F₆₄, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 64²-ovoid cap in AG(3,Â 64) [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
        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]

(68, 68+7, large)-Net in Base 8 — Upper bound on s

There is no (68, 75, large)-net in base 8, because

5 times m-reduction [i] would yield (68, 70, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]