MinT - Best Known (61, 85, s)-Nets in Base 7

Best Known (61, 85, s)-Nets in Base 7

(61, 85, 221)-Net over F₇ — Constructive and digital

Digital (61, 85, 221)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 13, 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, 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 (24, 48, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 24, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(61, 85, 2435)-Net over F₇ — Digital

Digital (61, 85, 2435)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸⁵, 2435, F₇, 24) (dual of [2435, 2350, 25]-code), using
- 26 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 5 times 0, 1, 17 times 0) [i] based on linear OA(7⁸¹, 2405, F₇, 24) (dual of [2405, 2324, 25]-code), using
  - constructionÂ X applied to C_e(23) âŠ‚ C_e(22) [i] based on
    1. linear OA(7⁸¹, 2401, F₇, 24) (dual of [2401, 2320, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    2. linear OA(7⁷⁷, 2401, F₇, 23) (dual of [2401, 2324, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(61, 85, 853723)-Net in Base 7 — Upper bound on s

There is no (61, 85, 853724)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 681292 660191 838341 029053 178058 767929 550447 636952 579710 301275 605418 358529 > 7⁸⁵ [i]