MinT - Best Known (66−13, 66, s)-Nets in Base 7

Best Known (66−13, 66, s)-Nets in Base 7

(66−13, 66, 2822)-Net over F₇ — Constructive and digital

Digital (53, 66, 2822)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 21)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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, 7, 13)-net over F₇, using
      - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (43, 56, 2801)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁵⁶, 2801, F₇, 13, 13) (dual of [(2801, 13), 36357, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(7⁵⁶, 16807, F₇, 13) (dual of [16807, 16751, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(66−13, 66, 39203)-Net over F₇ — Digital

Digital (53, 66, 39203)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−13, 66, large)-Net in Base 7 — Upper bound on s

There is no (53, 66, large)-net in base 7, because

11 times m-reduction [i] would yield (53, 55, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]