MinT - Best Known (39−5, 39, s)-Nets in Base 27

Best Known (39−5, 39, s)-Nets in Base 27

(39−5, 39, large)-Net over F₂₇ — Constructive and digital

Digital (34, 39, large)-net over F₂₇, using

27¹⁰ times duplication [i] based on digital (24, 29, 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 (18, 23, 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 (16, 21, 4194301)-net over F₂₇, using
        net defined by OOA [i] based on linear OOA(27²¹, 4194301, F₂₇, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(27²¹, large, F₂₇, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(39−5, 39, large)-Net in Base 27 — Upper bound on s

There is no (34, 39, large)-net in base 27, because

3 times m-reduction [i] would yield (34, 36, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]