MinT - Best Known (31−4, 31, s)-Nets in Base 25

Best Known (31−4, 31, s)-Nets in Base 25

(31−4, 31, large)-Net over F₂₅ — Constructive and digital

Digital (27, 31, large)-net over F₂₅, using

25² times duplication [i] based on digital (25, 29, large)-net over F₂₅, using
- t-expansion [i] based on digital (24, 29, large)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 6, large)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
    2. digital (18, 23, 4194327)-net over F₂₅, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 26)-net over F₂₅, using
        kÂ =Â 2 construction [i]
        digital (16, 21, 4194301)-net over F₂₅, using
        net defined by OOA [i] based on linear OOA(25²¹, 4194301, F₂₅, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(25²¹, large, F₂₅, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(31−4, 31, large)-Net in Base 25 — Upper bound on s

There is no (27, 31, large)-net in base 25, because

2 times m-reduction [i] would yield (27, 29, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]