Best Known (115, 115+14, s)-Nets in Base 3
(115, 115+14, 683283)-Net over F3 — Constructive and digital
Digital (115, 129, 683283)-net over F3, using
- 32 times duplication [i] based on digital (113, 127, 683283)-net over F3, using
- net defined by OOA [i] based on linear OOA(3127, 683283, F3, 14, 14) (dual of [(683283, 14), 9565835, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3127, 4782981, F3, 14) (dual of [4782981, 4782854, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 4782983, F3, 14) (dual of [4782983, 4782856, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 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(3127, 4782983, F3, 14) (dual of [4782983, 4782856, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3127, 4782981, F3, 14) (dual of [4782981, 4782854, 15]-code), using
- net defined by OOA [i] based on linear OOA(3127, 683283, F3, 14, 14) (dual of [(683283, 14), 9565835, 15]-NRT-code), using
(115, 115+14, 1594328)-Net over F3 — Digital
Digital (115, 129, 1594328)-net over F3, using
- 31 times duplication [i] based on digital (114, 128, 1594328)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 1594328, F3, 3, 14) (dual of [(1594328, 3), 4782856, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3128, 4782984, F3, 14) (dual of [4782984, 4782856, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3128, 4782985, F3, 14) (dual of [4782985, 4782857, 15]-code), using
- construction X4 applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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 X4 applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3128, 4782985, F3, 14) (dual of [4782985, 4782857, 15]-code), using
- OOA 3-folding [i] based on linear OA(3128, 4782984, F3, 14) (dual of [4782984, 4782856, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3128, 1594328, F3, 3, 14) (dual of [(1594328, 3), 4782856, 15]-NRT-code), using
(115, 115+14, large)-Net in Base 3 — Upper bound on s
There is no (115, 129, large)-net in base 3, because
- 12 times m-reduction [i] would yield (115, 117, large)-net in base 3, but