MinT - Best Known (82, 82+22, s)-Nets in Base 7

Best Known (82, 82+22, s)-Nets in Base 7

(82, 82+22, 1541)-Net over F₇ — Constructive and digital

Digital (82, 104, 1541)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 13)-net over F₇, using
  - 1 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (70, 92, 1528)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁹², 1528, F₇, 22, 22) (dual of [(1528, 22), 33524, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(7⁹², 16808, F₇, 22) (dual of [16808, 16716, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁹², 16813, F₇, 22) (dual of [16813, 16721, 23]-code), using
        constructionÂ X applied to C_e(21) âŠ‚ C_e(19) [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]
        linear OA(7⁸⁶, 16807, F₇, 20) (dual of [16807, 16721, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(7¹, 6, F₇, 1) (dual of [6, 5, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(7¹, s, F₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(82, 82+22, 22171)-Net over F₇ — Digital

Digital (82, 104, 22171)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+22, large)-Net in Base 7 — Upper bound on s

There is no (82, 104, large)-net in base 7, because

20 times m-reduction [i] would yield (82, 84, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]