MinT - Best Known (82, 82+12, s)-Nets in Base 9

Best Known (82, 82+12, s)-Nets in Base 9

(82, 82+12, 2796201)-Net over F₉ — Constructive and digital

Digital (82, 94, 2796201)-net over F₉, using

9¹ times duplication [i] based on digital (81, 93, 2796201)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁹³, 2796201, F₉, 15, 12) (dual of [(2796201, 15), 41942922, 13]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(9⁹³, 5592403, F₉, 3, 12) (dual of [(5592403, 3), 16777116, 13]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(9⁹⁰, 5592402, F₉, 3, 12) (dual of [(5592402, 3), 16777116, 13]-NRT-code), using
      - trace code [i] based on linear OOA(81⁴⁵, 2796201, F₈₁, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
        OOA 3-folding [i] based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(82, 82+12, large)-Net over F₉ — Digital

Digital (82, 94, large)-net over F₉, using

9⁵ times duplication [i] based on digital (77, 89, large)-net over F₉, using
- t-expansion [i] based on digital (76, 89, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(82, 82+12, large)-Net in Base 9 — Upper bound on s

There is no (82, 94, large)-net in base 9, because

10 times m-reduction [i] would yield (82, 84, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]