Best Known (228−17, 228, s)-Nets in Base 3
(228−17, 228, 1195767)-Net over F3 — Constructive and digital
Digital (211, 228, 1195767)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 22)-net over F3, using
- digital (197, 214, 1195745)-net over F3, using
- net defined by OOA [i] based on linear OOA(3214, 1195745, F3, 18, 17) (dual of [(1195745, 18), 21523196, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3214, 4782981, F3, 2, 17) (dual of [(4782981, 2), 9565748, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 4782984, F3, 2, 17) (dual of [(4782984, 2), 9565754, 18]-NRT-code), using
- trace code [i] based on linear OOA(9107, 2391492, F9, 2, 17) (dual of [(2391492, 2), 4782877, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9107, 4782984, F9, 17) (dual of [4782984, 4782877, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- 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(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 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(16) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(9107, 4782984, F9, 17) (dual of [4782984, 4782877, 18]-code), using
- trace code [i] based on linear OOA(9107, 2391492, F9, 2, 17) (dual of [(2391492, 2), 4782877, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 4782984, F3, 2, 17) (dual of [(4782984, 2), 9565754, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3214, 4782981, F3, 2, 17) (dual of [(4782981, 2), 9565748, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3214, 1195745, F3, 18, 17) (dual of [(1195745, 18), 21523196, 18]-NRT-code), using
(228−17, 228, large)-Net over F3 — Digital
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 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
(228−17, 228, large)-Net in Base 3 — Upper bound on s
There is no (211, 228, large)-net in base 3, because
- 15 times m-reduction [i] would yield (211, 213, large)-net in base 3, but