MinT - Best Known (70, 70+28, s)-Nets in Base 9

Best Known (70, 70+28, s)-Nets in Base 9

(70, 70+28, 740)-Net over F₉ — Constructive and digital

Digital (70, 98, 740)-net over F₉, using

10 times m-reduction [i] based on digital (70, 108, 740)-net over F₉, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(70, 70+28, 4774)-Net over F₉ — Digital

Digital (70, 98, 4774)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁹⁸, 4774, F₉, 28) (dual of [4774, 4676, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁹⁸, 6566, F₉, 28) (dual of [6566, 6468, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. linear OA(9⁹⁷, 6561, F₉, 28) (dual of [6561, 6464, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(9⁹³, 6561, F₉, 26) (dual of [6561, 6468, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(9¹, 5, F₉, 1) (dual of [5, 4, 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]

(70, 70+28, 3614633)-Net in Base 9 — Upper bound on s

There is no (70, 98, 3614634)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3279 185264 666541 219491 164348 343525 536276 725248 393376 948254 162575 100577 448540 775723 131919 628705 > 9⁹⁸ [i]