MinT - Best Known (55, 59, s)-Nets in Base 9

Best Known (55, 59, s)-Nets in Base 9

(55, 59, large)-Net over F₉ — Constructive and digital

Digital (55, 59, large)-net over F₉, using

3 times m-reduction [i] based on digital (55, 62, large)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 12, 5580100)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9¹², 5580100, F₉, 3, 3) (dual of [(5580100, 3), 16740288, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(9¹², 5580100, F₉, 2, 3) (dual of [(5580100, 2), 11160188, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9¹², 5580100, F₉, 3) (dual of [5580100, 5580088, 4]-code or 5580100-cap in PG(11,9)), using
        product of projective cap with cap in AG(5,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 (43, 50, 5592401)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁵⁰, 5592401, F₉, 9, 7) (dual of [(5592401, 9), 50331559, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(9⁵⁰, 5592402, F₉, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
        trace code [i] based on linear OOA(81²⁵, 2796201, F₈₁, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81²⁵, large, F₈₁, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(55, 59, large)-Net in Base 9 — Upper bound on s

There is no (55, 59, large)-net in base 9, because

2 times m-reduction [i] would yield (55, 57, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (55, 59, s)-Nets in Base 9

(55, 59, large)-Net over F9 — Constructive and digital

(55, 59, large)-Net in Base 9 — Upper bound on s

(55, 59, large)-Net over F₉ — Constructive and digital