MinT - Best Known (40−14, 40, s)-Nets in Base 7

Best Known (40−14, 40, s)-Nets in Base 7

(40−14, 40, 126)-Net over F₇ — Constructive and digital

Digital (26, 40, 126)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 50)-net over F₇, using
      - net defined by OOA [i] based on linear OOA(7⁴, 50, F₇, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(7⁴, 50, F₇, 2, 3) (dual of [(50, 2), 96, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁴, 50, F₇, 3) (dual of [50, 46, 4]-code or 50-cap in PG(3,7)), using
        ovoid in PG(3,Â 7) [i]
    2. digital (1, 8, 13)-net over F₇, using
      - 5 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]

(40−14, 40, 397)-Net over F₇ — Digital

Digital (26, 40, 397)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁴⁰, 397, F₇, 14) (dual of [397, 357, 15]-code), using
- 50 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0, 1, 30 times 0) [i] based on linear OA(7³⁶, 343, F₇, 14) (dual of [343, 307, 15]-code), using
  - 1 times truncation [i] based on linear OA(7³⁷, 344, F₇, 15) (dual of [344, 307, 16]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 344Â | 7⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(40−14, 40, 38005)-Net in Base 7 — Upper bound on s

There is no (26, 40, 38006)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 6366 879357 396681 738194 641634 674657 > 7⁴⁰ [i]