MinT - Best Known (57−5, 57, s)-Nets in Base 27

Best Known (57−5, 57, s)-Nets in Base 27

(57−5, 57, large)-Net over F₂₇ — Constructive and digital

Digital (52, 57, large)-net over F₂₇, using

27⁸ times duplication [i] based on digital (44, 49, large)-net over F₂₇, using
- t-expansion [i] based on digital (42, 49, large)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (6, 9, large)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁹, large, F₂₇, 3, 3), using
        appending kth column [i] based on linear OOA(27⁹, large, F₂₇, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁹, large, F₂₇, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
    2. digital (33, 40, 5592400)-net over F₂₇, using
      - (u,Â u+v)-construction [i] based on
        digital (6, 9, large)-net over F₂₇ (see above)
        digital (24, 31, 2796200)-net over F₂₇, using
        net defined by OOA [i] based on linear OOA(27³¹, 2796200, F₂₇, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(27³¹, 8388601, F₂₇, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(27³¹, large, F₂₇, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(57−5, 57, large)-Net in Base 27 — Upper bound on s

There is no (52, 57, large)-net in base 27, because

3 times m-reduction [i] would yield (52, 54, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (57−5, 57, s)-Nets in Base 27

(57−5, 57, large)-Net over F27 — Constructive and digital

(57−5, 57, large)-Net in Base 27 — Upper bound on s

(57−5, 57, large)-Net over F₂₇ — Constructive and digital