MinT - Best Known (67, 67+10, s)-Nets in Base 9

Best Known (67, 67+10, s)-Nets in Base 9

(67, 67+10, 3355440)-Net over F₉ — Constructive and digital

Digital (67, 77, 3355440)-net over F₉, using

9³ times duplication [i] based on digital (64, 74, 3355440)-net over F₉, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81³⁷, 1677720, F₈₁, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(81³⁷, 8388600, F₈₁, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(81³⁷, large, F₈₁, 10) (dual of [large, large−37, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(67, 67+10, large)-Net over F₉ — Digital

Digital (67, 77, large)-net over F₉, using

9⁴ times duplication [i] based on digital (63, 73, large)-net over F₉, using
- t-expansion [i] based on digital (62, 73, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁷³, large, F₉, 11) (dual of [large, large−73, 12]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(67, 67+10, large)-Net in Base 9 — Upper bound on s

There is no (67, 77, large)-net in base 9, because

8 times m-reduction [i] would yield (67, 69, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]