MinT - Best Known (59−4, 59, s)-Nets in Base 8

Best Known (59−4, 59, s)-Nets in Base 8

(59−4, 59, large)-Net over F₈ — Constructive and digital

Digital (55, 59, large)-net over F₈, using

3 times m-reduction [i] based on digital (55, 62, large)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 12, 2931457)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8¹², 2931457, F₈, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(8¹², 2931457, F₈, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8¹², 2931457, F₈, 3) (dual of [2931457, 2931445, 4]-code or 2931457-cap in PG(11,8)), using
        product of projective cap with cap in AG(5,Â 8) [i] based on linear OA(8⁷, 4224, F₈, 3) (dual of [4224, 4217, 4]-code or 4224-cap in PG(6,8)), using
        product of two caps with tangent hyperplane [i]
  2. digital (43, 50, 5592401)-net over F₈, using
    - net defined by OOA [i] based on linear OOA(8⁵⁰, 5592401, F₈, 9, 7) (dual of [(5592401, 9), 50331559, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on 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]

(59−4, 59, large)-Net in Base 8 — Upper bound on s

There is no (55, 59, large)-net in base 8, because

2 times m-reduction [i] would yield (55, 57, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (59−4, 59, s)-Nets in Base 8

(59−4, 59, large)-Net over F8 — Constructive and digital

(59−4, 59, large)-Net in Base 8 — Upper bound on s

(59−4, 59, large)-Net over F₈ — Constructive and digital