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

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

(53, 53+6, large)-Net over F₉ — Constructive and digital

Digital (53, 59, large)-net over F₉, using

9³ times duplication [i] based on digital (50, 56, large)-net over F₉, using
- trace code for nets [i] based on digital (22, 28, 5592401)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81²⁸, 5592401, F₈₁, 9, 6) (dual of [(5592401, 9), 50331581, 7]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(81²⁸, 5592402, F₈₁, 3, 6) (dual of [(5592402, 3), 16777178, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(81⁷, large, F₈₁, 3, 3), using
        appending kth column [i] based on linear OOA(81⁷, large, F₈₁, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁷, large, F₈₁, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 81²-ovoid cap in AG(3,Â 81) [i] based on linear OA(81⁴, 6562, F₈₁, 3) (dual of [6562, 6558, 4]-code or 6562-cap in PG(3,81)), using
        ovoid in PG(3,Â 81) [i]
        linear OOA(81²¹, 2796201, F₈₁, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
        OOA 3-folding [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, 53+6, large)-Net in Base 9 — Upper bound on s

There is no (53, 59, large)-net in base 9, because

4 times m-reduction [i] would yield (53, 55, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]