MinT - Best Known (35−11, 35, s)-Nets in Base 9

Best Known (35−11, 35, s)-Nets in Base 9

(35−11, 35, 364)-Net over F₉ — Constructive and digital

Digital (24, 35, 364)-net over F₉, using

9¹ times duplication [i] based on digital (23, 34, 364)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (13, 24, 200)-net over F₉, using
    - trace code for nets [i] based on digital (1, 12, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]

(35−11, 35, 400)-Net in Base 9 — Constructive

(24, 35, 400)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (6, 11, 351)-net in base 9, using
  - net defined by OOA [i] based on OOA(9¹¹, 351, S₉, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(9¹¹, 703, S₉, 5), using
      - discarding parts of the base [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (13, 24, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 12, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]

(35−11, 35, 1248)-Net over F₉ — Digital

Digital (24, 35, 1248)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³⁵, 1248, F₉, 11) (dual of [1248, 1213, 12]-code), using
- 509 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) [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]

(35−11, 35, 1003677)-Net in Base 9 — Upper bound on s

There is no (24, 35, 1003678)-net in base 9, because

1 times m-reduction [i] would yield (24, 34, 1003678)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 278 129075 411437 259402 326510 027569 > 9³⁴ [i]