MinT - Best Known (121−13, 121, s)-Nets in Base 9

Best Known (121−13, 121, s)-Nets in Base 9

(121−13, 121, 2800575)-Net over F₉ — Constructive and digital

Digital (108, 121, 2800575)-net over F₉, using

9¹ times duplication [i] based on digital (107, 120, 2800575)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 22, 4375)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9²², 4375, F₉, 6, 6) (dual of [(4375, 6), 26228, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(9²², 13125, F₉, 6) (dual of [13125, 13103, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(9²², 13126, F₉, 6) (dual of [13126, 13104, 7]-code), using
        trace code [i] based on linear OA(81¹¹, 6563, F₈₁, 6) (dual of [6563, 6552, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹¹, 6561, F₈₁, 6) (dual of [6561, 6550, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(81⁹, 6561, F₈₁, 5) (dual of [6561, 6552, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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]
  2. digital (85, 98, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁸, 2796200, F₉, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁸, 8388601, F₉, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁸, 8388602, F₉, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁹, 4194301, F₈₁, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁹, 8388602, F₈₁, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(121−13, 121, large)-Net over F₉ — Digital

Digital (108, 121, large)-net over F₉, using

t-expansion [i] based on digital (107, 121, large)-net over F₉, using
- 4 times m-reduction [i] based on digital (107, 125, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(121−13, 121, large)-Net in Base 9 — Upper bound on s

There is no (108, 121, large)-net in base 9, because

11 times m-reduction [i] would yield (108, 110, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (121−13, 121, s)-Nets in Base 9

(121−13, 121, 2800575)-Net over F9 — Constructive and digital

(121−13, 121, large)-Net over F9 — Digital

(121−13, 121, large)-Net in Base 9 — Upper bound on s

(121−13, 121, 2800575)-Net over F₉ — Constructive and digital

(121−13, 121, large)-Net over F₉ — Digital