MinT - Best Known (78−3, 78, s)-Nets in Base 5

Best Known (78−3, 78, s)-Nets in Base 5

(78−3, 78, large)-Net over F₅ — Constructive and digital

Digital (75, 78, large)-net over F₅, using

5¹ times duplication [i] based on digital (74, 77, large)-net over F₅, using
- t-expansion [i] based on digital (70, 77, large)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (12, 15, 3050326)-net over F₅, using
      - net defined by OOA [i] based on linear OOA(5¹⁵, 3050326, F₅, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(5¹⁵, 3050326, F₅, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5¹⁵, 3050326, F₅, 3) (dual of [3050326, 3050311, 4]-code or 3050326-cap in PG(14,5)), using
        product of projective cap with cap in AG(6,Â 5) [i] based on linear OA(5⁹, 4700, F₅, 3) (dual of [4700, 4691, 4]-code or 4700-cap in PG(8,5)), using
        caps completed using a computer search [i]
    2. digital (55, 62, 5592400)-net over F₅, using
      - trace code for nets [i] based on digital (24, 31, 2796200)-net over F₂₅, using
        net defined by OOA [i] based on linear OOA(25³¹, 2796200, F₂₅, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(25³¹, 8388601, F₂₅, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(25³¹, large, F₂₅, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(78−3, 78, large)-Net in Base 5 — Upper bound on s

There is no (75, 78, large)-net in base 5, because

1 times m-reduction [i] would yield (75, 77, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (78−3, 78, s)-Nets in Base 5

(78−3, 78, large)-Net over F5 — Constructive and digital

(78−3, 78, large)-Net in Base 5 — Upper bound on s

(78−3, 78, large)-Net over F₅ — Constructive and digital