MinT - Best Known (40, 40+23, s)-Nets in Base 7

Best Known (40, 40+23, s)-Nets in Base 7

(40, 40+23, 121)-Net over F₇ — Constructive and digital

Digital (40, 63, 121)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 21)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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, 12, 13)-net over F₇, using
      - 1 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (23, 46, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 23, 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]

(40, 40+23, 410)-Net over F₇ — Digital

Digital (40, 63, 410)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁶³, 410, F₇, 23) (dual of [410, 347, 24]-code), using
- 59 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 27 times 0) [i] based on linear OA(7⁵⁸, 346, F₇, 23) (dual of [346, 288, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
    1. linear OA(7⁵⁸, 343, F₇, 23) (dual of [343, 285, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(7⁵⁵, 343, F₇, 22) (dual of [343, 288, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(7⁰, 3, F₇, 0) (dual of [3, 3, 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]

(40, 40+23, 47432)-Net in Base 7 — Upper bound on s

There is no (40, 63, 47433)-net in base 7, because

1 times m-reduction [i] would yield (40, 62, 47433)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 24893 313200 628231 952800 920361 429971 571787 088793 615127 > 7⁶² [i]