MinT - Best Known (83−21, 83, s)-Nets in Base 9

Best Known (83−21, 83, s)-Nets in Base 9

(83−21, 83, 1312)-Net over F₉ — Constructive and digital

Digital (62, 83, 1312)-net over F₉, using

9¹ times duplication [i] based on digital (61, 82, 1312)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁸², 1312, F₉, 21, 21) (dual of [(1312, 21), 27470, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(9⁸², 13121, F₉, 21) (dual of [13121, 13039, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁸², 13124, F₉, 21) (dual of [13124, 13042, 22]-code), using
      - trace code [i] based on linear OA(81⁴¹, 6562, F₈₁, 21) (dual of [6562, 6521, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(83−21, 83, 13004)-Net over F₉ — Digital

Digital (62, 83, 13004)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸³, 13004, F₉, 21) (dual of [13004, 12921, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁸³, 13127, F₉, 21) (dual of [13127, 13044, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(9⁸², 13126, F₉, 21) (dual of [13126, 13044, 22]-code), using
    - trace code [i] based on linear OA(81⁴¹, 6563, F₈₁, 21) (dual of [6563, 6522, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(81⁴¹, 6561, F₈₁, 21) (dual of [6561, 6520, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(81³⁹, 6561, F₈₁, 20) (dual of [6561, 6522, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(81⁰, 2, F₈₁, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(83−21, 83, large)-Net in Base 9 — Upper bound on s

There is no (62, 83, large)-net in base 9, because

19 times m-reduction [i] would yield (62, 64, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]