MinT - Best Known (56−4, 56, s)-Nets in Base 8

Best Known (56−4, 56, s)-Nets in Base 8

(56−4, 56, large)-Net over F₈ — Constructive and digital

Digital (52, 56, large)-net over F₈, using

t-expansion [i] based on digital (50, 56, large)-net over F₈, using
- trace code for nets [i] based on digital (22, 28, 5592401)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁸, 5592401, F₆₄, 9, 6) (dual of [(5592401, 9), 50331581, 7]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(64²⁸, 5592402, F₆₄, 3, 6) (dual of [(5592402, 3), 16777178, 7]-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(64²¹, 2796201, F₆₄, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(56−4, 56, large)-Net in Base 8 — Upper bound on s

There is no (52, 56, large)-net in base 8, because

2 times m-reduction [i] would yield (52, 54, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]