MinT - Best Known (85−25, 85, s)-Nets in Base 7

Best Known (85−25, 85, s)-Nets in Base 7

(85−25, 85, 215)-Net over F₇ — Constructive and digital

Digital (60, 85, 215)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 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 (27, 52, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 26, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(85−25, 85, 1904)-Net over F₇ — Digital

Digital (60, 85, 1904)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁵, 1904, F₇, 25) (dual of [1904, 1819, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁸⁵, 2401, F₇, 25) (dual of [2401, 2316, 26]-code), using
  - an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(85−25, 85, 725925)-Net in Base 7 — Upper bound on s

There is no (60, 85, 725926)-net in base 7, because

1 times m-reduction [i] would yield (60, 84, 725926)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 97328 516060 199549 983600 057877 331700 116098 797078 067737 151990 551659 130041 > 7⁸⁴ [i]