MinT - Best Known (231−17, 231, s)-Nets in Base 3

Best Known (231−17, 231, s)-Nets in Base 3

(231−17, 231, 1195775)-Net over F₃ — Constructive and digital

Digital (214, 231, 1195775)-net over F₃, using

3¹ times duplication [i] based on digital (213, 230, 1195775)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 18, 32)-net over F₃, using
    - trace code for nets [i] based on digital (1, 9, 16)-net over F₉, using
      - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
        a function field by GarcÃa and Quoos [i]
  2. digital (195, 212, 1195743)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3²¹², 1195743, F₃, 18, 17) (dual of [(1195743, 18), 21523162, 18]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(3²¹², 4782973, F₃, 2, 17) (dual of [(4782973, 2), 9565734, 18]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(3²¹², 4782976, F₃, 2, 17) (dual of [(4782976, 2), 9565740, 18]-NRT-code), using
        trace code [i] based on linear OOA(9¹⁰⁶, 2391488, F₉, 2, 17) (dual of [(2391488, 2), 4782870, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(9¹⁰⁶, 4782976, F₉, 17) (dual of [4782976, 4782870, 18]-code), using
        constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
        linear OA(9¹⁰⁶, 4782969, F₉, 17) (dual of [4782969, 4782863, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(9⁹⁹, 4782969, F₉, 16) (dual of [4782969, 4782870, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 9⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(9⁰, 7, F₉, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(231−17, 231, large)-Net over F₃ — Digital

Digital (214, 231, large)-net over F₃, using

3³ times duplication [i] based on digital (211, 228, large)-net over F₃, using
- t-expansion [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]

(231−17, 231, large)-Net in Base 3 — Upper bound on s

There is no (214, 231, large)-net in base 3, because

15 times m-reduction [i] would yield (214, 216, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (231−17, 231, s)-Nets in Base 3

(231−17, 231, 1195775)-Net over F3 — Constructive and digital

(231−17, 231, large)-Net over F3 — Digital

(231−17, 231, large)-Net in Base 3 — Upper bound on s

(231−17, 231, 1195775)-Net over F₃ — Constructive and digital

(231−17, 231, large)-Net over F₃ — Digital