MinT - Best Known (41−7, 41, s)-Nets in Base 7

Best Known (41−7, 41, s)-Nets in Base 7

(41−7, 41, 78442)-Net over F₇ — Constructive and digital

Digital (34, 41, 78442)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 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 (31, 38, 78434)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7³⁸, 78434, F₇, 7, 7) (dual of [(78434, 7), 549000, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(7³⁸, 235303, F₇, 7) (dual of [235303, 235265, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(7³⁸, 235304, F₇, 7) (dual of [235304, 235266, 8]-code), using
        trace code [i] based on linear OA(49¹⁹, 117652, F₄₉, 7) (dual of [117652, 117633, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(49¹⁹, 117649, F₄₉, 7) (dual of [117649, 117630, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(49¹⁶, 117649, F₄₉, 6) (dual of [117649, 117633, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(49⁰, 3, F₄₉, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(41−7, 41, 297106)-Net over F₇ — Digital

Digital (34, 41, 297106)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−7, 41, large)-Net in Base 7 — Upper bound on s

There is no (34, 41, large)-net in base 7, because

5 times m-reduction [i] would yield (34, 36, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (41−7, 41, s)-Nets in Base 7

(41−7, 41, 78442)-Net over F7 — Constructive and digital

(41−7, 41, 297106)-Net over F7 — Digital

(41−7, 41, large)-Net in Base 7 — Upper bound on s

(41−7, 41, 78442)-Net over F₇ — Constructive and digital

(41−7, 41, 297106)-Net over F₇ — Digital