MinT - Best Known (45−6, 45, s)-Nets in Base 27

Best Known (45−6, 45, s)-Nets in Base 27

(45−6, 45, large)-Net over F₂₇ — Constructive and digital

Digital (39, 45, large)-net over F₂₇, using

27¹ times duplication [i] based on digital (38, 44, large)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 9, large)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁹, large, F₂₇, 3, 3), using
      - appending kth column [i] based on linear OOA(27⁹, large, F₂₇, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁹, large, F₂₇, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
  2. digital (29, 35, 5592402)-net over F₂₇, using
    - (u,Â u+v)-construction [i] based on
      1. digital (6, 9, large)-net over F₂₇ (see above)
      2. digital (20, 26, 2796201)-net over F₂₇, using
        net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(45−6, 45, large)-Net in Base 27 — Upper bound on s

There is no (39, 45, large)-net in base 27, because

4 times m-reduction [i] would yield (39, 41, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]