MinT - Best Known (111−10, 111, s)-Nets in Base 3

Best Known (111−10, 111, s)-Nets in Base 3

(111−10, 111, 1678090)-Net over F₃ — Constructive and digital

Digital (101, 111, 1678090)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 20, 370)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁰, 370, F₃, 5, 5) (dual of [(370, 5), 1830, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(3²⁰, 741, F₃, 5) (dual of [741, 721, 6]-code), using
      - constructionÂ XX applied to C₁Â =Â C([361,364]), C₂Â =Â C([363,365]), C₃Â =Â C₁Â +Â C₂Â =Â C([363,364]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([361,365]) [i] based on
        linear OA(3¹³, 728, F₃, 4) (dual of [728, 715, 5]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {361,362,363,364}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3¹³, 728, F₃, 3) (dual of [728, 715, 4]-code or 728-cap in PG(12,3)), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {363,364,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(3¹⁹, 728, F₃, 5) (dual of [728, 709, 6]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {361,362,363,364,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(3⁷, 728, F₃, 2) (dual of [728, 721, 3]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {363,364}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 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]
        linear OA(3⁰, 6, F₃, 0) (dual of [6, 6, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 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 (81, 91, 1677720)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹¹, 1677720, F₃, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(3⁹¹, 8388600, F₃, 10) (dual of [8388600, 8388509, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹¹, large, F₃, 10) (dual of [large, large−91, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(111−10, 111, 6840327)-Net over F₃ — Digital

Digital (101, 111, 6840327)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹¹, 6840327, F₃, 10) (dual of [6840327, 6840216, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹¹, large, F₃, 10) (dual of [large, large−111, 11]-code), using
  - 20 times code embedding in larger space [i] based on linear OA(3⁹¹, large, F₃, 10) (dual of [large, large−91, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(111−10, 111, large)-Net in Base 3 — Upper bound on s

There is no (101, 111, large)-net in base 3, because

8 times m-reduction [i] would yield (101, 103, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (111−10, 111, s)-Nets in Base 3

(111−10, 111, 1678090)-Net over F3 — Constructive and digital

(111−10, 111, 6840327)-Net over F3 — Digital

(111−10, 111, large)-Net in Base 3 — Upper bound on s

(111−10, 111, 1678090)-Net over F₃ — Constructive and digital

(111−10, 111, 6840327)-Net over F₃ — Digital