MinT - Best Known (123, 123+18, s)-Nets in Base 9

Best Known (123, 123+18, s)-Nets in Base 9

(123, 123+18, 1864134)-Net over F₉ — Constructive and digital

Digital (123, 141, 1864134)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁴¹, 1864134, F₉, 21, 18) (dual of [(1864134, 21), 39146673, 19]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(9¹⁴¹, 5592403, F₉, 3, 18) (dual of [(5592403, 3), 16777068, 19]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(9¹³⁸, 5592402, F₉, 3, 18) (dual of [(5592402, 3), 16777068, 19]-NRT-code), using
    - trace code [i] based on linear OOA(81⁶⁹, 2796201, F₈₁, 3, 18) (dual of [(2796201, 3), 8388534, 19]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(81⁶⁹, large, F₈₁, 18) (dual of [large, large−69, 19]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(123, 123+18, large)-Net over F₉ — Digital

Digital (123, 141, large)-net over F₉, using

9⁴ times duplication [i] based on digital (119, 137, large)-net over F₉, using
- t-expansion [i] based on digital (117, 137, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹³⁷, large, F₉, 20) (dual of [large, large−137, 21]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(123, 123+18, large)-Net in Base 9 — Upper bound on s

There is no (123, 141, large)-net in base 9, because

16 times m-reduction [i] would yield (123, 125, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]