MinT - Best Known (123−16, 123, s)-Nets in Base 9

Best Known (123−16, 123, s)-Nets in Base 9

(123−16, 123, 2097150)-Net over F₉ — Constructive and digital

Digital (107, 123, 2097150)-net over F₉, using

9¹ times duplication [i] based on digital (106, 122, 2097150)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9¹²², 2097150, F₉, 18, 16) (dual of [(2097150, 18), 37748578, 17]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OOA(9¹²², 8388601, F₉, 2, 16) (dual of [(8388601, 2), 16777080, 17]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(9¹²², 8388602, F₉, 2, 16) (dual of [(8388602, 2), 16777082, 17]-NRT-code), using
      - trace code [i] based on linear OOA(81⁶¹, 4194301, F₈₁, 2, 16) (dual of [(4194301, 2), 8388541, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁶¹, 8388602, F₈₁, 16) (dual of [8388602, 8388541, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁶¹, large, F₈₁, 16) (dual of [large, large−61, 17]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(123−16, 123, large)-Net over F₉ — Digital

Digital (107, 123, large)-net over F₉, using

2 times m-reduction [i] based on digital (107, 125, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(123−16, 123, large)-Net in Base 9 — Upper bound on s

There is no (107, 123, large)-net in base 9, because

14 times m-reduction [i] would yield (107, 109, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]