Best Known (128, 128+23, s)-Nets in Base 3
(128, 128+23, 5368)-Net over F3 — Constructive and digital
Digital (128, 151, 5368)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 5368, F3, 23, 23) (dual of [(5368, 23), 123313, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 11-folding and stacking with additional row [i] based on linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using
(128, 128+23, 19686)-Net over F3 — Digital
Digital (128, 151, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3151, 19686, F3, 3, 23) (dual of [(19686, 3), 58907, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3151, 59058, F3, 23) (dual of [59058, 58907, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3151, 59059, F3, 23) (dual of [59059, 58908, 24]-code), using
- OOA 3-folding [i] based on linear OA(3151, 59058, F3, 23) (dual of [59058, 58907, 24]-code), using
(128, 128+23, 7873766)-Net in Base 3 — Upper bound on s
There is no (128, 151, 7873767)-net in base 3, because
- 1 times m-reduction [i] would yield (128, 150, 7873767)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 369988 950440 764283 019421 702430 639616 236954 779733 659765 160588 860736 436235 > 3150 [i]