MinT - Best Known (76, 76+19, s)-Nets in Base 7

Best Known (76, 76+19, s)-Nets in Base 7

(76, 76+19, 1888)-Net over F₇ — Constructive and digital

Digital (76, 95, 1888)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (5, 14, 21)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 10, 13)-net over F₇, using
      - 3 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (62, 81, 1867)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁸¹, 1867, F₇, 19, 19) (dual of [(1867, 19), 35392, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(7⁸¹, 16804, F₇, 19) (dual of [16804, 16723, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁸¹, 16807, F₇, 19) (dual of [16807, 16726, 20]-code), using
        an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(76, 76+19, 36335)-Net over F₇ — Digital

Digital (76, 95, 36335)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(76, 76+19, large)-Net in Base 7 — Upper bound on s

There is no (76, 95, large)-net in base 7, because

17 times m-reduction [i] would yield (76, 78, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]