Best Known (177, 177+22, s)-Nets in Base 3
(177, 177+22, 434816)-Net over F3 — Constructive and digital
Digital (177, 199, 434816)-net over F3, using
- 31 times duplication [i] based on digital (176, 198, 434816)-net over F3, using
- net defined by OOA [i] based on linear OOA(3198, 434816, F3, 22, 22) (dual of [(434816, 22), 9565754, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3198, 4782976, F3, 22) (dual of [4782976, 4782778, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 4782984, F3, 22) (dual of [4782984, 4782786, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3198, 4782984, F3, 22) (dual of [4782984, 4782786, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3198, 4782976, F3, 22) (dual of [4782976, 4782778, 23]-code), using
- net defined by OOA [i] based on linear OOA(3198, 434816, F3, 22, 22) (dual of [(434816, 22), 9565754, 23]-NRT-code), using
(177, 177+22, 1066032)-Net over F3 — Digital
Digital (177, 199, 1066032)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3199, 1066032, F3, 4, 22) (dual of [(1066032, 4), 4263929, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3199, 1195746, F3, 4, 22) (dual of [(1195746, 4), 4782785, 23]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3198, 1195746, F3, 4, 22) (dual of [(1195746, 4), 4782786, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3198, 4782984, F3, 22) (dual of [4782984, 4782786, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 X applied to Ce(21) ⊂ Ce(19) [i] based on
- OOA 4-folding [i] based on linear OA(3198, 4782984, F3, 22) (dual of [4782984, 4782786, 23]-code), using
- 31 times duplication [i] based on linear OOA(3198, 1195746, F3, 4, 22) (dual of [(1195746, 4), 4782786, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3199, 1195746, F3, 4, 22) (dual of [(1195746, 4), 4782785, 23]-NRT-code), using
(177, 177+22, large)-Net in Base 3 — Upper bound on s
There is no (177, 199, large)-net in base 3, because
- 20 times m-reduction [i] would yield (177, 179, large)-net in base 3, but