Best Known (183−31, 183, s)-Nets in Base 3
(183−31, 183, 1312)-Net over F3 — Constructive and digital
Digital (152, 183, 1312)-net over F3, using
- 32 times duplication [i] based on digital (150, 181, 1312)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 1312, F3, 31, 31) (dual of [(1312, 31), 40491, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3181, 19681, F3, 31) (dual of [19681, 19500, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3181, 19681, F3, 31) (dual of [19681, 19500, 32]-code), using
- net defined by OOA [i] based on linear OOA(3181, 1312, F3, 31, 31) (dual of [(1312, 31), 40491, 32]-NRT-code), using
(183−31, 183, 6831)-Net over F3 — Digital
Digital (152, 183, 6831)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3183, 6831, F3, 2, 31) (dual of [(6831, 2), 13479, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3183, 9847, F3, 2, 31) (dual of [(9847, 2), 19511, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3183, 19694, F3, 31) (dual of [19694, 19511, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3182, 19693, F3, 31) (dual of [19693, 19511, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(30) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3182, 19693, F3, 31) (dual of [19693, 19511, 32]-code), using
- OOA 2-folding [i] based on linear OA(3183, 19694, F3, 31) (dual of [19694, 19511, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3183, 9847, F3, 2, 31) (dual of [(9847, 2), 19511, 32]-NRT-code), using
(183−31, 183, 1976080)-Net in Base 3 — Upper bound on s
There is no (152, 183, 1976081)-net in base 3, because
- 1 times m-reduction [i] would yield (152, 182, 1976081)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 685 597556 978909 260008 779436 201838 958253 006681 252223 048223 879453 399241 336132 280758 182571 > 3182 [i]