MinT - Best Known (23−3, 23, s)-Nets in Base 27

Best Known (23−3, 23, s)-Nets in Base 27

(23−3, 23, large)-Net over F₂₇ — Constructive and digital

Digital (20, 23, large)-net over F₂₇, using

1 times m-reduction [i] based on digital (20, 24, large)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 6, large)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
  2. digital (14, 18, 4194329)-net over F₂₇, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 2, 28)-net over F₂₇, using
        kÂ =Â 2 construction [i]
      2. digital (12, 16, 4194301)-net over F₂₇, using
        net defined by OOA [i] based on linear OOA(27¹⁶, 4194301, F₂₇, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(27¹⁶, 8388602, F₂₇, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹⁶, large, F₂₇, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(23−3, 23, large)-Net in Base 27 — Upper bound on s

There is no (20, 23, large)-net in base 27, because

1 times m-reduction [i] would yield (20, 22, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]