MinT - Best Known (53, 53+8, s)-Nets in Base 9

Best Known (53, 53+8, s)-Nets in Base 9

(53, 53+8, 4194301)-Net over F₉ — Constructive and digital

Digital (53, 61, 4194301)-net over F₉, using

9¹ times duplication [i] based on digital (52, 60, 4194301)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁶⁰, 4194301, F₉, 10, 8) (dual of [(4194301, 10), 41942950, 9]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(9⁶⁰, large, F₉, 2, 8), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(9⁵⁸, 8388602, F₉, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
      - trace code [i] based on linear OOA(81²⁹, 4194301, F₈₁, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81²⁹, 8388602, F₈₁, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(81²⁹, large, F₈₁, 8) (dual of [large, large−29, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(53, 53+8, large)-Net over F₉ — Digital

Digital (53, 61, large)-net over F₉, using

t-expansion [i] based on digital (52, 61, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+8, large)-Net in Base 9 — Upper bound on s

There is no (53, 61, large)-net in base 9, because

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