MinT - Best Known (53−6, 53, s)-Nets in Base 9

Best Known (53−6, 53, s)-Nets in Base 9

(53−6, 53, 7004232)-Net over F₉ — Constructive and digital

Digital (47, 53, 7004232)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 11, 1411830)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹¹, 1411830, F₉, 3, 3) (dual of [(1411830, 3), 4235479, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(9¹¹, 1411830, F₉, 2, 3) (dual of [(1411830, 2), 2823649, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9¹¹, 1411830, F₉, 3) (dual of [1411830, 1411819, 4]-code or 1411830-cap in PG(10,9)), using
        product of projective cap with cap in AG(4,Â 9) [i] based on linear OA(9⁷, 6723, F₉, 3) (dual of [6723, 6716, 4]-code or 6723-cap in PG(6,9)), using
        product of two caps with tangent hyperplane [i]
2. digital (36, 42, 5592402)-net over F₉, using
  - trace code for nets [i] based on digital (15, 21, 2796201)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81²¹, 2796201, F₈₁, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(81²¹, large, F₈₁, 6) (dual of [large, large−21, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(53−6, 53, large)-Net over F₉ — Digital

Digital (47, 53, large)-net over F₉, using

t-expansion [i] based on digital (46, 53, large)-net over F₉, using
- 1 times m-reduction [i] based on digital (46, 54, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(53−6, 53, large)-Net in Base 9 — Upper bound on s

There is no (47, 53, large)-net in base 9, because

4 times m-reduction [i] would yield (47, 49, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]