Best Known (134, 134+21, s)-Nets in Base 3
(134, 134+21, 17715)-Net over F3 — Constructive and digital
Digital (134, 155, 17715)-net over F3, using
- net defined by OOA [i] based on linear OOA(3155, 17715, F3, 21, 21) (dual of [(17715, 21), 371860, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3155, 177151, F3, 21) (dual of [177151, 176996, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 177158, F3, 21) (dual of [177158, 177003, 22]-code), using
- 1 times truncation [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- 1 times truncation [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 177158, F3, 21) (dual of [177158, 177003, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3155, 177151, F3, 21) (dual of [177151, 176996, 22]-code), using
(134, 134+21, 59052)-Net over F3 — Digital
Digital (134, 155, 59052)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3155, 59052, F3, 3, 21) (dual of [(59052, 3), 177001, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3155, 177156, F3, 21) (dual of [177156, 177001, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 177158, F3, 21) (dual of [177158, 177003, 22]-code), using
- 1 times truncation [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- 1 times truncation [i] based on linear OA(3156, 177159, F3, 22) (dual of [177159, 177003, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 177158, F3, 21) (dual of [177158, 177003, 22]-code), using
- OOA 3-folding [i] based on linear OA(3155, 177156, F3, 21) (dual of [177156, 177001, 22]-code), using
(134, 134+21, large)-Net in Base 3 — Upper bound on s
There is no (134, 155, large)-net in base 3, because
- 19 times m-reduction [i] would yield (134, 136, large)-net in base 3, but