MinT - Best Known (68, 97, s)-Nets in Base 7

Best Known (68, 97, s)-Nets in Base 7

(68, 97, 213)-Net over F₇ — Constructive and digital

Digital (68, 97, 213)-net over F₇, using

7¹ times duplication [i] based on digital (67, 96, 213)-net over F₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. 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 (14, 28, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 14, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
        Niederreiter sequence [i]
  3. digital (29, 58, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 29, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(68, 97, 1824)-Net over F₇ — Digital

Digital (68, 97, 1824)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁷, 1824, F₇, 29) (dual of [1824, 1727, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁹⁷, 2401, F₇, 29) (dual of [2401, 2304, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(68, 97, 628432)-Net in Base 7 — Upper bound on s

There is no (68, 97, 628433)-net in base 7, because

1 times m-reduction [i] would yield (68, 96, 628433)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1347 163692 358169 641352 849542 773826 237393 983433 661907 189285 210604 219495 473437 635057 > 7⁹⁶ [i]