MinT - Best Known (28−7, 28, s)-Nets in Base 64

Best Known (28−7, 28, s)-Nets in Base 64

(28−7, 28, 2796265)-Net over F₆₄ — Constructive and digital

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
    1. 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]
    2. 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]

(28−7, 28, large)-Net over F₆₄ — Digital

Digital (21, 28, large)-net over F₆₄, using

1 times m-reduction [i] based on digital (21, 29, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(28−7, 28, large)-Net in Base 64 — Upper bound on s

There is no (21, 28, large)-net in base 64, because

5 times m-reduction [i] would yield (21, 23, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]