MinT - Best Known (86, 86+21, s)-Nets in Base 7

Best Known (86, 86+21, s)-Nets in Base 7

(86, 86+21, 1706)-Net over F₇ — Constructive and digital

Digital (86, 107, 1706)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 6, 13)-net over F₇, using
      - 7 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
    2. digital (1, 11, 13)-net over F₇, using
      - 2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇ (see above)
2. digital (69, 90, 1680)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁹⁰, 1680, F₇, 21, 21) (dual of [(1680, 21), 35190, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(7⁹⁰, 16801, F₇, 21) (dual of [16801, 16711, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁹⁰, 16806, F₇, 21) (dual of [16806, 16716, 22]-code), using
        1 times truncation [i] based on linear OA(7⁹¹, 16807, F₇, 22) (dual of [16807, 16716, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(86, 86+21, 45975)-Net over F₇ — Digital

Digital (86, 107, 45975)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+21, large)-Net in Base 7 — Upper bound on s

There is no (86, 107, large)-net in base 7, because

19 times m-reduction [i] would yield (86, 88, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]