MinT - Best Known (108−10, 108, s)-Nets in Base 9

Best Known (108−10, 108, s)-Nets in Base 9

(108−10, 108, 6710880)-Net over F₉ — Constructive and digital

Digital (98, 108, 6710880)-net over F₉, using

9¹ times duplication [i] based on digital (97, 107, 6710880)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 33, 4194301)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9³³, 4194301, F₉, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(9³³, large, F₉, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
  2. 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]

(108−10, 108, large)-Net over F₉ — Digital

Digital (98, 108, large)-net over F₉, using

t-expansion [i] based on digital (95, 108, large)-net over F₉, using
- 3 times m-reduction [i] based on digital (95, 111, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(108−10, 108, large)-Net in Base 9 — Upper bound on s

There is no (98, 108, large)-net in base 9, because

8 times m-reduction [i] would yield (98, 100, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]