MinT - Best Known (111, 123, s)-Nets in Base 9

Best Known (111, 123, s)-Nets in Base 9

(111, 123, 3150496)-Net over F₉ — Constructive and digital

Digital (111, 123, 3150496)-net over F₉, using

9¹ times duplication [i] based on digital (110, 122, 3150496)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (26, 32, 354296)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9³², 354296, F₉, 6, 6) (dual of [(354296, 6), 2125744, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(9³², 1062888, F₉, 6) (dual of [1062888, 1062856, 7]-code), using
        trace code [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 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 (78, 90, 2796200)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9⁹⁰, 2796200, F₉, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁰, 8388601, F₉, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(9⁹⁰, 8388602, F₉, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁵, 4194301, F₈₁, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁵, 8388602, F₈₁, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(111, 123, large)-Net over F₉ — Digital

Digital (111, 123, large)-net over F₉, using

t-expansion [i] based on digital (110, 123, large)-net over F₉, using
- 6 times m-reduction [i] based on digital (110, 129, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹²⁹, large, F₉, 19) (dual of [large, large−129, 20]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(111, 123, large)-Net in Base 9 — Upper bound on s

There is no (111, 123, large)-net in base 9, because

10 times m-reduction [i] would yield (111, 113, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111, 123, s)-Nets in Base 9

(111, 123, 3150496)-Net over F9 — Constructive and digital

(111, 123, large)-Net over F9 — Digital

(111, 123, large)-Net in Base 9 — Upper bound on s

(111, 123, 3150496)-Net over F₉ — Constructive and digital

(111, 123, large)-Net over F₉ — Digital