Best Known (223−18, 223, s)-Nets in Base 3
(223−18, 223, 932615)-Net over F3 — Constructive and digital
Digital (205, 223, 932615)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (34, 43, 548)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 548, F3, 9, 9) (dual of [(548, 9), 4889, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
- net defined by OOA [i] based on linear OOA(343, 548, F3, 9, 9) (dual of [(548, 9), 4889, 10]-NRT-code), using
- digital (162, 180, 932067)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- net defined by OOA [i] based on linear OOA(3180, 932067, F3, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
- digital (34, 43, 548)-net over F3, using
(223−18, 223, large)-Net over F3 — Digital
Digital (205, 223, large)-net over F3, using
- 37 times duplication [i] based on digital (198, 216, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
(223−18, 223, large)-Net in Base 3 — Upper bound on s
There is no (205, 223, large)-net in base 3, because
- 16 times m-reduction [i] would yield (205, 207, large)-net in base 3, but