MinT - Best Known (27, 27+6, s)-Nets in Base 64

Best Known (27, 27+6, s)-Nets in Base 64

(27, 27+6, large)-Net over F₆₄ — Constructive and digital

Digital (27, 33, large)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 5, 2796201)-net over F₆₄, using
  - s-reduction based on digital (3, 5, large)-net over F₆₄, using
    - kÂ =Â 2 construction [i]
2. digital (4, 7, 2796201)-net over F₆₄, using
  - s-reduction based on digital (4, 7, large)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁷, large, F₆₄, 3, 3), using
      - appending kth column [i] based on linear OOA(64⁷, large, F₆₄, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 64²-ovoid cap in AG(3,Â 64) [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]
3. 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]

(27, 27+6, large)-Net in Base 64 — Upper bound on s

There is no (27, 33, large)-net in base 64, because

4 times m-reduction [i] would yield (27, 29, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]