MinT - Best Known (150−13, 150, s)-Nets in Base 9

Best Known (150−13, 150, s)-Nets in Base 9

(150−13, 150, 5595640)-Net over F₉ — Constructive and digital

Digital (137, 150, 5595640)-net over F₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 11, 3240)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹¹, 3240, F₉, 4, 4) (dual of [(3240, 4), 12949, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(9¹¹, 3240, F₉, 3, 4) (dual of [(3240, 3), 9709, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(9¹¹, 6480, F₉, 4) (dual of [6480, 6469, 5]-code), using
        base reduction for projective spaces (embedding PG(5,81) in PG(10,9)) [i] based on linear OA(81⁶, 6480, F₈₁, 4) (dual of [6480, 6474, 5]-code), using
        1 times truncation [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (35, 41, 2796200)-net over F₉, using
  - s-reduction based on digital (35, 41, 2796201)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁴¹, 2796201, F₉, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(9⁴¹, large, F₉, 6) (dual of [large, large−41, 7]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. 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]

(150−13, 150, large)-Net over F₉ — Digital

Digital (137, 150, large)-net over F₉, using

9⁵ times duplication [i] based on digital (132, 145, large)-net over F₉, using
- t-expansion [i] based on digital (124, 145, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(150−13, 150, large)-Net in Base 9 — Upper bound on s

There is no (137, 150, large)-net in base 9, because

11 times m-reduction [i] would yield (137, 139, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (150−13, 150, s)-Nets in Base 9

(150−13, 150, 5595640)-Net over F9 — Constructive and digital

(150−13, 150, large)-Net over F9 — Digital

(150−13, 150, large)-Net in Base 9 — Upper bound on s

(150−13, 150, 5595640)-Net over F₉ — Constructive and digital

(150−13, 150, large)-Net over F₉ — Digital