MinT - Best Known (23, 28, s)-Nets in Base 27

Best Known (23, 28, s)-Nets in Base 27

(23, 28, large)-Net over F₂₇ — Constructive and digital

Digital (23, 28, large)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 4194301)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (4, 6, 4194301)-net over F₂₇, using
  - s-reduction based on digital (4, 6, large)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
3. 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]

(23, 28, large)-Net in Base 27 — Upper bound on s

There is no (23, 28, large)-net in base 27, because

3 times m-reduction [i] would yield (23, 25, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]