MinT - Best Known (82−4, 82, s)-Nets in Base 4

Best Known (82−4, 82, s)-Nets in Base 4

(82−4, 82, large)-Net over F₄ — Constructive and digital

Digital (78, 82, large)-net over F₄, using

t-expansion [i] based on digital (77, 82, large)-net over F₄, using
- 2 times m-reduction [i] based on digital (77, 84, large)-net over F₄, using
  - trace code for nets [i] based on digital (21, 28, 2796265)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²⁸, 2796265, F₆₄, 9, 7) (dual of [(2796265, 9), 25166357, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(64²⁸, 2796266, F₆₄, 3, 7) (dual of [(2796266, 3), 8388770, 8]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(64³, 65, F₆₄, 3, 3) (dual of [(65, 3), 192, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;192,64) [i]
        linear OOA(64²⁵, 2796201, F₆₄, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(82−4, 82, large)-Net in Base 4 — Upper bound on s

There is no (78, 82, large)-net in base 4, because

2 times m-reduction [i] would yield (78, 80, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]