MinT - Best Known (51−3, 51, s)-Nets in Base 5

Best Known (51−3, 51, s)-Nets in Base 5

(51−3, 51, large)-Net over F₅ — Constructive and digital

Digital (48, 51, large)-net over F₅, using

5⁵ times duplication [i] based on digital (43, 46, large)-net over F₅, using
- t-expansion [i] based on digital (41, 46, large)-net over F₅, using
  - trace code for nets [i] based on digital (18, 23, 4194326)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²³, 4194326, F₂₅, 6, 5) (dual of [(4194326, 6), 25165933, 6]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(25²³, 4194327, F₂₅, 2, 5) (dual of [(4194327, 2), 8388631, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(25², 26, F₂₅, 2, 2) (dual of [(26, 2), 50, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;50,25) [i]
        linear OOA(25²¹, 4194301, F₂₅, 2, 5) (dual of [(4194301, 2), 8388581, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25²¹, 8388602, F₂₅, 5) (dual of [8388602, 8388581, 6]-code), using
        discarding factors / shortening the dual code 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]

(51−3, 51, large)-Net in Base 5 — Upper bound on s

There is no (48, 51, large)-net in base 5, because

1 times m-reduction [i] would yield (48, 50, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]