MinT - Best Known (26−10, 26, s)-Nets in Base 9

Best Known (26−10, 26, s)-Nets in Base 9

(26−10, 26, 232)-Net over F₉ — Constructive and digital

Digital (16, 26, 232)-net over F₉, using

2 times m-reduction [i] based on digital (16, 28, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 14, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(26−10, 26, 448)-Net over F₉ — Digital

Digital (16, 26, 448)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9²⁶, 448, F₉, 10) (dual of [448, 422, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(9²⁶, 733, F₉, 10) (dual of [733, 707, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
    1. linear OA(9²⁵, 729, F₉, 10) (dual of [729, 704, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(9²², 729, F₉, 8) (dual of [729, 707, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(9¹, 4, F₉, 1) (dual of [4, 3, 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]

(26−10, 26, 29837)-Net in Base 9 — Upper bound on s

There is no (16, 26, 29838)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 461126 710277 848328 690865 > 9²⁶ [i]