MinT - Best Known (50−6, 50, s)-Nets in Base 32

Best Known (50−6, 50, s)-Nets in Base 32

(50−6, 50, large)-Net over F₃₂ — Constructive and digital

Digital (44, 50, large)-net over F₃₂, using

32³ times duplication [i] based on digital (41, 47, large)-net over F₃₂, using
- t-expansion [i] based on digital (40, 47, large)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (5, 8, 3214400)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁸, 3214400, F₃₂, 3, 3) (dual of [(3214400, 3), 9643192, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(32⁸, 3214400, F₃₂, 2, 3) (dual of [(3214400, 2), 6428792, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁸, 3214400, F₃₂, 3) (dual of [3214400, 3214392, 4]-code or 3214400-cap in PG(7,32)), using
        product of projective cap with cap in AG(4,Â 32) [i] based on linear OA(32⁴, 1025, F₃₂, 3) (dual of [1025, 1021, 4]-code or 1025-cap in PG(3,32)), using
        ovoid in PG(3,Â 32) [i]
    2. digital (32, 39, 5592400)-net over F₃₂, using
      - (u,Â u+v)-construction [i] based on
        digital (5, 8, 3214400)-net over F₃₂ (see above)
        digital (24, 31, 2796200)-net over F₃₂, using
        net defined by OOA [i] based on linear OOA(32³¹, 2796200, F₃₂, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(32³¹, 8388601, F₃₂, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(32³¹, large, F₃₂, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(50−6, 50, large)-Net in Base 32 — Upper bound on s

There is no (44, 50, large)-net in base 32, because

4 times m-reduction [i] would yield (44, 46, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (50−6, 50, s)-Nets in Base 32

(50−6, 50, large)-Net over F32 — Constructive and digital

(50−6, 50, large)-Net in Base 32 — Upper bound on s

(50−6, 50, large)-Net over F₃₂ — Constructive and digital