MinT - Best Known (105−12, 105, s)-Nets in Base 3

Best Known (105−12, 105, s)-Nets in Base 3

(105−12, 105, 265722)-Net over F₃ — Constructive and digital

Digital (93, 105, 265722)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁰⁵, 265722, F₃, 12, 12) (dual of [(265722, 12), 3188559, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3¹⁰⁵, 1594332, F₃, 12) (dual of [1594332, 1594227, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁰⁵, 1594337, F₃, 12) (dual of [1594337, 1594232, 13]-code), using
    - 1 times truncation [i] based on linear OA(3¹⁰⁶, 1594338, F₃, 13) (dual of [1594338, 1594232, 14]-code), using
      - constructionÂ X4 applied to C_e(12) âŠ‚ C_e(10) [i] based on
        linear OA(3¹⁰⁵, 1594323, F₃, 13) (dual of [1594323, 1594218, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁹², 1594323, F₃, 11) (dual of [1594323, 1594231, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
        dual of repetition code with length 15 [i]
        linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 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]

(105−12, 105, 598554)-Net over F₃ — Digital

Digital (93, 105, 598554)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁰⁵, 598554, F₃, 2, 12) (dual of [(598554, 2), 1197003, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁰⁵, 797168, F₃, 2, 12) (dual of [(797168, 2), 1594231, 13]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁰⁵, 1594336, F₃, 12) (dual of [1594336, 1594231, 13]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁰⁵, 1594337, F₃, 12) (dual of [1594337, 1594232, 13]-code), using
      - 1 times truncation [i] based on linear OA(3¹⁰⁶, 1594338, F₃, 13) (dual of [1594338, 1594232, 14]-code), using
        constructionÂ X4 applied to C_e(12) âŠ‚ C_e(10) [i] based on
        linear OA(3¹⁰⁵, 1594323, F₃, 13) (dual of [1594323, 1594218, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(3⁹², 1594323, F₃, 11) (dual of [1594323, 1594231, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
        dual of repetition code with length 15 [i]
        linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 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]

(105−12, 105, large)-Net in Base 3 — Upper bound on s

There is no (93, 105, large)-net in base 3, because

10 times m-reduction [i] would yield (93, 95, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (105−12, 105, s)-Nets in Base 3

(105−12, 105, 265722)-Net over F3 — Constructive and digital

(105−12, 105, 598554)-Net over F3 — Digital

(105−12, 105, large)-Net in Base 3 — Upper bound on s

(105−12, 105, 265722)-Net over F₃ — Constructive and digital

(105−12, 105, 598554)-Net over F₃ — Digital