MinT - Best Known (53, 82, s)-Nets in Base 9

Best Known (53, 82, s)-Nets in Base 9

(53, 82, 344)-Net over F₉ — Constructive and digital

Digital (53, 82, 344)-net over F₉, using

10 times m-reduction [i] based on digital (53, 92, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(53, 82, 895)-Net over F₉ — Digital

Digital (53, 82, 895)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸², 895, F₉, 29) (dual of [895, 813, 30]-code), using
- 157 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 11 times 0, 1, 28 times 0, 1, 48 times 0, 1, 62 times 0) [i] based on linear OA(9⁷⁶, 732, F₉, 29) (dual of [732, 656, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(9⁷⁶, 729, F₉, 29) (dual of [729, 653, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(9⁷³, 729, F₉, 28) (dual of [729, 656, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 9³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [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]

(53, 82, 250800)-Net in Base 9 — Upper bound on s

There is no (53, 82, 250801)-net in base 9, because

1 times m-reduction [i] would yield (53, 81, 250801)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 196635 151367 670371 496886 670871 183426 707524 922463 753754 049798 012580 175690 322481 > 9⁸¹ [i]