MinT - Best Known (81, 89, s)-Nets in Base 7

Best Known (81, 89, s)-Nets in Base 7

(81, 89, 6291450)-Net over F₇ — Constructive and digital

Digital (81, 89, 6291450)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 9, 2097150)-net over F₇, using
  - s-reduction based on digital (7, 9, 6725601)-net over F₇, using
    - kÂ =Â 2 construction [i]
2. digital (21, 25, 2097150)-net over F₇, using
  - s-reduction based on digital (21, 25, 2882404)-net over F₇, using
    - net defined by OOA [i] based on linear OOA(7²⁵, 2882404, F₇, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(7²⁵, 2882404, F₇, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(7²⁵, 5764808, F₇, 4) (dual of [5764808, 5764783, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(7²⁵, 5764809, F₇, 4) (dual of [5764809, 5764784, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(7²⁵, 5764801, F₇, 4) (dual of [5764801, 5764776, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(7¹⁷, 5764801, F₇, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(7⁰, 8, F₇, 0) (dual of [8, 8, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
3. digital (47, 55, 2097150)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁵⁵, 2097150, F₇, 8, 8) (dual of [(2097150, 8), 16777145, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(7⁵⁵, 8388600, F₇, 8) (dual of [8388600, 8388545, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁵⁵, large, F₇, 8) (dual of [large, large−55, 9]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 20176803Â | 7⁹âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(81, 89, large)-Net over F₇ — Digital

Digital (81, 89, large)-net over F₇, using

t-expansion [i] based on digital (78, 89, large)-net over F₇, using
- 1 times m-reduction [i] based on digital (78, 90, large)-net over F₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁰, large, F₇, 12) (dual of [large, large−90, 13]-code), using
    - trace code [i] based on linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using
      - an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(81, 89, large)-Net in Base 7 — Upper bound on s

There is no (81, 89, large)-net in base 7, because

6 times m-reduction [i] would yield (81, 83, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (81, 89, s)-Nets in Base 7

(81, 89, 6291450)-Net over F7 — Constructive and digital

(81, 89, large)-Net over F7 — Digital

(81, 89, large)-Net in Base 7 — Upper bound on s

(81, 89, 6291450)-Net over F₇ — Constructive and digital

(81, 89, large)-Net over F₇ — Digital