Best Known (227−18, 227, s)-Nets in Base 3
(227−18, 227, 1062883)-Net over F3 — Constructive and digital
Digital (209, 227, 1062883)-net over F3, using
- 1 times m-reduction [i] based on digital (209, 228, 1062883)-net over F3, using
- net defined by OOA [i] based on linear OOA(3228, 1062883, F3, 21, 19) (dual of [(1062883, 21), 22320315, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3228, 3188650, F3, 3, 19) (dual of [(3188650, 3), 9565722, 20]-NRT-code), using
- trace code [i] based on linear OOA(9114, 1594325, F9, 3, 19) (dual of [(1594325, 3), 4782861, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(9114, 4782975, F9, 19) (dual of [4782975, 4782861, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9114, 4782977, F9, 19) (dual of [4782977, 4782863, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(91, 8, F9, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(9114, 4782977, F9, 19) (dual of [4782977, 4782863, 20]-code), using
- OOA 3-folding [i] based on linear OA(9114, 4782975, F9, 19) (dual of [4782975, 4782861, 20]-code), using
- trace code [i] based on linear OOA(9114, 1594325, F9, 3, 19) (dual of [(1594325, 3), 4782861, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3228, 3188650, F3, 3, 19) (dual of [(3188650, 3), 9565722, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3228, 1062883, F3, 21, 19) (dual of [(1062883, 21), 22320315, 20]-NRT-code), using
(227−18, 227, large)-Net over F3 — Digital
Digital (209, 227, large)-net over F3, using
- 1 times m-reduction [i] based on digital (209, 228, large)-net over F3, using
- embedding of OOA with Gilbert–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 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
(227−18, 227, large)-Net in Base 3 — Upper bound on s
There is no (209, 227, large)-net in base 3, because
- 16 times m-reduction [i] would yield (209, 211, large)-net in base 3, but