MinT - Best Known (67, 94, s)-Nets in Base 7

Best Known (67, 94, s)-Nets in Base 7

(67, 94, 221)-Net over F₇ — Constructive and digital

Digital (67, 94, 221)-net over F₇, using

generalized (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 (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 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, 54, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 27, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(67, 94, 2346)-Net over F₇ — Digital

Digital (67, 94, 2346)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁴, 2346, F₇, 27) (dual of [2346, 2252, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁹⁴, 2410, F₇, 27) (dual of [2410, 2316, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(24) [i] based on
    1. linear OA(7⁹³, 2401, F₇, 27) (dual of [2401, 2308, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. 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]
    3. linear OA(7¹, 9, F₇, 1) (dual of [9, 8, 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]

(67, 94, 1049428)-Net in Base 7 — Upper bound on s

There is no (67, 94, 1049429)-net in base 7, because

1 times m-reduction [i] would yield (67, 93, 1049429)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 927531 453993 994717 731212 009836 947162 970697 823764 465302 352412 021716 892837 902527 > 7⁹³ [i]