Best Known (134−19, 134, s)-Nets in Base 3
(134−19, 134, 19684)-Net over F3 — Constructive and digital
Digital (115, 134, 19684)-net over F3, using
- net defined by OOA [i] based on linear OOA(3134, 19684, F3, 19, 19) (dual of [(19684, 19), 373862, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3134, 177157, F3, 19) (dual of [177157, 177023, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3134, 177159, F3, 19) (dual of [177159, 177025, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3134, 177159, F3, 19) (dual of [177159, 177025, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3134, 177157, F3, 19) (dual of [177157, 177023, 20]-code), using
(134−19, 134, 47148)-Net over F3 — Digital
Digital (115, 134, 47148)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3134, 47148, F3, 3, 19) (dual of [(47148, 3), 141310, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3134, 59053, F3, 3, 19) (dual of [(59053, 3), 177025, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3134, 177159, F3, 19) (dual of [177159, 177025, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3134, 177159, F3, 19) (dual of [177159, 177025, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3134, 59053, F3, 3, 19) (dual of [(59053, 3), 177025, 20]-NRT-code), using
(134−19, 134, large)-Net in Base 3 — Upper bound on s
There is no (115, 134, large)-net in base 3, because
- 17 times m-reduction [i] would yield (115, 117, large)-net in base 3, but