MinT - Best Known (35−13, 35, s)-Nets in Base 9

Best Known (35−13, 35, s)-Nets in Base 9

(35−13, 35, 300)-Net over F₉ — Constructive and digital

Digital (22, 35, 300)-net over F₉, using

1 times m-reduction [i] based on digital (22, 36, 300)-net over F₉, using
- trace code for nets [i] based on digital (4, 18, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(35−13, 35, 541)-Net over F₉ — Digital

Digital (22, 35, 541)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³⁵, 541, F₉, 13) (dual of [541, 506, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(9³⁵, 736, F₉, 13) (dual of [736, 701, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [i] based on
    1. 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]
    2. linear OA(9²⁸, 729, F₉, 11) (dual of [729, 701, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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]

(35−13, 35, 95607)-Net in Base 9 — Upper bound on s

There is no (22, 35, 95608)-net in base 9, because

1 times m-reduction [i] would yield (22, 34, 95608)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 278 138687 598688 734188 743959 027585 > 9³⁴ [i]