MinT - Best Known (223−12, 223, s)-Nets in Base 3

Best Known (223−12, 223, s)-Nets in Base 3

(223−12, 223, 5612090)-Net over F₃ — Constructive and digital

Digital (211, 223, 5612090)-net over F₃, using

3¹ times duplication [i] based on digital (210, 222, 5612090)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (36, 42, 19690)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3⁴², 19690, F₃, 6, 6) (dual of [(19690, 6), 118098, 7]-NRT-code), using
      - appending kth column [i] based on linear OOA(3⁴², 19690, F₃, 5, 6) (dual of [(19690, 5), 98408, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(3⁴², 59070, F₃, 6) (dual of [59070, 59028, 7]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(3) [i] based on
        linear OA(3⁴¹, 59049, F₃, 7) (dual of [59049, 59008, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3²¹, 59049, F₃, 4) (dual of [59049, 59028, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(3¹, 21, F₃, 1) (dual of [21, 20, 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 (168, 180, 5592400)-net over F₃, using
    - trace code for nets [i] based on 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]

(223−12, 223, large)-Net over F₃ — Digital

Digital (211, 223, large)-net over F₃, using

t-expansion [i] based on digital (209, 223, large)-net over F₃, using
- 5 times m-reduction [i] based on digital (209, 228, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, large, F₃, 19) (dual of [large, large−228, 20]-code), using
    - 47 times code embedding in larger space [i] based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(223−12, 223, large)-Net in Base 3 — Upper bound on s

There is no (211, 223, large)-net in base 3, because

10 times m-reduction [i] would yield (211, 213, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (223−12, 223, s)-Nets in Base 3

(223−12, 223, 5612090)-Net over F3 — Constructive and digital

(223−12, 223, large)-Net over F3 — Digital

(223−12, 223, large)-Net in Base 3 — Upper bound on s

(223−12, 223, 5612090)-Net over F₃ — Constructive and digital

(223−12, 223, large)-Net over F₃ — Digital