MinT - Best Known (222, 222+13, s)-Nets in Base 3

Best Known (222, 222+13, s)-Nets in Base 3

(222, 222+13, 5598968)-Net over F₃ — Constructive and digital

Digital (222, 235, 5598968)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (33, 39, 6568)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3³⁹, 6568, F₃, 6, 6) (dual of [(6568, 6), 39369, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3³⁹, 6568, F₃, 5, 6) (dual of [(6568, 5), 32801, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(3³⁹, 19704, F₃, 6) (dual of [19704, 19665, 7]-code), using
        1 times code embedding in larger space [i] based on linear OA(3³⁸, 19703, F₃, 6) (dual of [19703, 19665, 7]-code), using
        constructionÂ X4 applied to C_e(6) âŠ‚ C_e(3) [i] based on
        linear OA(3³⁷, 19683, F₃, 7) (dual of [19683, 19646, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3¹⁹, 19683, F₃, 4) (dual of [19683, 19664, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(3¹⁹, 20, F₃, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
        dual of repetition code with length 20 [i]
        linear OA(3¹, 20, F₃, 1) (dual of [20, 19, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (183, 196, 5592400)-net over F₃, using
  - trace code for nets [i] based on 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]

(222, 222+13, large)-Net over F₃ — Digital

Digital (222, 235, large)-net over F₃, using

t-expansion [i] based on digital (221, 235, large)-net over F₃, using
- 6 times m-reduction [i] based on digital (221, 241, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, large, F₃, 20) (dual of [large, large−241, 21]-code), using
    - strength reduction [i] based on linear OA(3²⁴¹, large, F₃, 25) (dual of [large, large−241, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(222, 222+13, large)-Net in Base 3 — Upper bound on s

There is no (222, 235, large)-net in base 3, because

11 times m-reduction [i] would yield (222, 224, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (222, 222+13, s)-Nets in Base 3

(222, 222+13, 5598968)-Net over F3 — Constructive and digital

(222, 222+13, large)-Net over F3 — Digital

(222, 222+13, large)-Net in Base 3 — Upper bound on s

(222, 222+13, 5598968)-Net over F₃ — Constructive and digital

(222, 222+13, large)-Net over F₃ — Digital