MinT - Best Known (19, 25, s)-Nets in Base 64

Best Known (19, 25, s)-Nets in Base 64

(19, 25, 2800298)-Net over F₆₄ — Constructive and digital

Digital (19, 25, 2800298)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 4, 4097)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴, 4097, F₆₄, 3, 3) (dual of [(4097, 3), 12287, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(64⁴, 4097, F₆₄, 2, 3) (dual of [(4097, 2), 8190, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
2. digital (15, 21, 2796201)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²¹, 2796201, F₆₄, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(19, 25, large)-Net over F₆₄ — Digital

Digital (19, 25, large)-net over F₆₄, using

t-expansion [i] based on digital (18, 25, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [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]

(19, 25, large)-Net in Base 64 — Upper bound on s

There is no (19, 25, large)-net in base 64, because

4 times m-reduction [i] would yield (19, 21, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]