Best Known (126−14, 126, s)-Nets in Base 3
(126−14, 126, 227769)-Net over F3 — Constructive and digital
Digital (112, 126, 227769)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (104, 118, 227762)-net over F3, using
- net defined by OOA [i] based on linear OOA(3118, 227762, F3, 14, 14) (dual of [(227762, 14), 3188550, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3118, 1594334, F3, 14) (dual of [1594334, 1594216, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3118, 1594336, F3, 14) (dual of [1594336, 1594218, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3118, 1594336, F3, 14) (dual of [1594336, 1594218, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3118, 1594334, F3, 14) (dual of [1594334, 1594216, 15]-code), using
- net defined by OOA [i] based on linear OOA(3118, 227762, F3, 14, 14) (dual of [(227762, 14), 3188550, 15]-NRT-code), using
- digital (1, 8, 7)-net over F3, using
(126−14, 126, 586725)-Net over F3 — Digital
Digital (112, 126, 586725)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3126, 586725, F3, 2, 14) (dual of [(586725, 2), 1173324, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 797185, F3, 2, 14) (dual of [(797185, 2), 1594244, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3124, 797184, F3, 2, 14) (dual of [(797184, 2), 1594244, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3124, 1594368, F3, 14) (dual of [1594368, 1594244, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(3124, 1594368, F3, 14) (dual of [1594368, 1594244, 15]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3124, 797184, F3, 2, 14) (dual of [(797184, 2), 1594244, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 797185, F3, 2, 14) (dual of [(797185, 2), 1594244, 15]-NRT-code), using
(126−14, 126, large)-Net in Base 3 — Upper bound on s
There is no (112, 126, large)-net in base 3, because
- 12 times m-reduction [i] would yield (112, 114, large)-net in base 3, but