MinT - Best Known (25, 25+11, s)-Nets in Base 9

Best Known (25, 25+11, s)-Nets in Base 9

(25, 25+11, 410)-Net over F₉ — Constructive and digital

Digital (25, 36, 410)-net over F₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 4, 82)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁴, 82, F₉, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(9⁴, 82, F₉, 2, 3) (dual of [(82, 2), 160, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
        ovoid in PG(3,Â 9) [i]
2. digital (5, 10, 164)-net over F₉, using
  - trace code for nets [i] based on digital (0, 5, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
3. digital (11, 22, 164)-net over F₉, using
  - trace code for nets [i] based on digital (0, 11, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(25, 25+11, 1548)-Net over F₉ — Digital

Digital (25, 36, 1548)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³⁶, 1548, F₉, 11) (dual of [1548, 1512, 12]-code), using
- 808 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 11 times 0, 1, 32 times 0, 1, 79 times 0, 1, 151 times 0, 1, 228 times 0, 1, 298 times 0) [i] based on linear OA(9²⁸, 732, F₉, 11) (dual of [732, 704, 12]-code), using
  - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
    1. 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]
    2. 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]
    3. linear OA(9⁰, 3, F₉, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(25, 25+11, 1557553)-Net in Base 9 — Upper bound on s

There is no (25, 36, 1557554)-net in base 9, because

1 times m-reduction [i] would yield (25, 35, 1557554)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2503 155969 229226 112981 504989 952849 > 9³⁵ [i]