Best Known (226−17, 226, s)-Nets in Base 3
(226−17, 226, 1195765)-Net over F3 — Constructive and digital
Digital (209, 226, 1195765)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 22)-net over F3, using
- digital (195, 212, 1195743)-net over F3, using
- net defined by OOA [i] based on linear OOA(3212, 1195743, F3, 18, 17) (dual of [(1195743, 18), 21523162, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3212, 4782973, F3, 2, 17) (dual of [(4782973, 2), 9565734, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 4782976, F3, 2, 17) (dual of [(4782976, 2), 9565740, 18]-NRT-code), using
- trace code [i] based on linear OOA(9106, 2391488, F9, 2, 17) (dual of [(2391488, 2), 4782870, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9106, 4782976, F9, 17) (dual of [4782976, 4782870, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [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(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(9106, 4782976, F9, 17) (dual of [4782976, 4782870, 18]-code), using
- trace code [i] based on linear OOA(9106, 2391488, F9, 2, 17) (dual of [(2391488, 2), 4782870, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 4782976, F3, 2, 17) (dual of [(4782976, 2), 9565740, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3212, 4782973, F3, 2, 17) (dual of [(4782973, 2), 9565734, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3212, 1195743, F3, 18, 17) (dual of [(1195743, 18), 21523162, 18]-NRT-code), using
(226−17, 226, large)-Net over F3 — Digital
Digital (209, 226, large)-net over F3, using
- 2 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
(226−17, 226, large)-Net in Base 3 — Upper bound on s
There is no (209, 226, large)-net in base 3, because
- 15 times m-reduction [i] would yield (209, 211, large)-net in base 3, but