MinT - Best Known (64−5, 64, s)-Nets in Base 7

Best Known (64−5, 64, s)-Nets in Base 7

(64−5, 64, large)-Net over F₇ — Constructive and digital

Digital (59, 64, large)-net over F₇, using

1 times m-reduction [i] based on digital (59, 65, large)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 13, 6120101)-net over F₇, using
    - net defined by OOA [i] based on linear OOA(7¹³, 6120101, F₇, 3, 3) (dual of [(6120101, 3), 18360290, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(7¹³, 6120101, F₇, 2, 3) (dual of [(6120101, 2), 12240189, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7¹³, 6120101, F₇, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
        product of projective cap with cap in AG(6,Â 7) [i] based on linear OA(7⁷, 2499, F₇, 3) (dual of [2499, 2492, 4]-code or 2499-cap in PG(6,7)), using
        product of two caps with tangent hyperplane [i]
  2. digital (46, 52, 5592402)-net over F₇, using
    - trace code for nets [i] based on digital (20, 26, 2796201)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49²⁶, 2796201, F₄₉, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(49²⁶, large, F₄₉, 6) (dual of [large, large−26, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 25679568Â | 49⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(64−5, 64, large)-Net in Base 7 — Upper bound on s

There is no (59, 64, large)-net in base 7, because

3 times m-reduction [i] would yield (59, 61, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]