MinT - Best Known (41−15, 41, s)-Nets in Base 9

Best Known (41−15, 41, s)-Nets in Base 9

(41−15, 41, 320)-Net over F₉ — Constructive and digital

Digital (26, 41, 320)-net over F₉, using

1 times m-reduction [i] based on digital (26, 42, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 21, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(41−15, 41, 605)-Net over F₉ — Digital

Digital (26, 41, 605)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁴¹, 605, F₉, 15) (dual of [605, 564, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁴¹, 736, F₉, 15) (dual of [736, 695, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(12) [i] based on
    1. linear OA(9⁴⁰, 729, F₉, 15) (dual of [729, 689, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    2. linear OA(9³⁴, 729, F₉, 13) (dual of [729, 695, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(9¹, 7, F₉, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(41−15, 41, 119847)-Net in Base 9 — Upper bound on s

There is no (26, 41, 119848)-net in base 9, because

1 times m-reduction [i] would yield (26, 40, 119848)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 147 813193 519961 034825 779000 569091 467713 > 9⁴⁰ [i]