MinT - Best Known (81−11, 81, s)-Nets in Base 9

Best Known (81−11, 81, s)-Nets in Base 9

(81−11, 81, 1677756)-Net over F₉ — Constructive and digital

Digital (70, 81, 1677756)-net over F₉, using

9¹ times duplication [i] based on digital (69, 80, 1677756)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (2, 7, 36)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁷, 36, F₉, 5, 5) (dual of [(36, 5), 173, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(9⁷, 73, F₉, 5) (dual of [73, 66, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (62, 73, 1677720)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁷³, 1677720, F₉, 11, 11) (dual of [(1677720, 11), 18454847, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(9⁷³, 8388601, F₉, 11) (dual of [8388601, 8388528, 12]-code), using
        discarding factors / shortening the dual code 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]

(81−11, 81, large)-Net over F₉ — Digital

Digital (70, 81, large)-net over F₉, using

t-expansion [i] based on digital (69, 81, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸¹, large, F₉, 12) (dual of [large, large−81, 13]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(81−11, 81, large)-Net in Base 9 — Upper bound on s

There is no (70, 81, large)-net in base 9, because

9 times m-reduction [i] would yield (70, 72, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]