MinT - Best Known (111−10, 111, s)-Nets in Base 9

Best Known (111−10, 111, s)-Nets in Base 9

(111−10, 111, 6710962)-Net over F₉ — Constructive and digital

Digital (101, 111, 6710962)-net over F₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 4, 82)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁴, 82, F₉, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(9⁴, 82, F₉, 2, 3) (dual of [(82, 2), 160, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
        ovoid in PG(3,Â 9) [i]
2. digital (28, 33, 3355440)-net over F₉, using
  - s-reduction based on 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]
3. 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]

(111−10, 111, large)-Net over F₉ — Digital

Digital (101, 111, large)-net over F₉, using

7 times m-reduction [i] based on digital (101, 118, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(111−10, 111, large)-Net in Base 9 — Upper bound on s

There is no (101, 111, large)-net in base 9, because

8 times m-reduction [i] would yield (101, 103, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111−10, 111, s)-Nets in Base 9

(111−10, 111, 6710962)-Net over F9 — Constructive and digital

(111−10, 111, large)-Net over F9 — Digital

(111−10, 111, large)-Net in Base 9 — Upper bound on s

(111−10, 111, 6710962)-Net over F₉ — Constructive and digital

(111−10, 111, large)-Net over F₉ — Digital