Best Known (171−28, 171, s)-Nets in Base 3
(171−28, 171, 1408)-Net over F3 — Constructive and digital
Digital (143, 171, 1408)-net over F3, using
- net defined by OOA [i] based on linear OOA(3171, 1408, F3, 28, 28) (dual of [(1408, 28), 39253, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3171, 19712, F3, 28) (dual of [19712, 19541, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 19718, F3, 28) (dual of [19718, 19547, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3171, 19718, F3, 28) (dual of [19718, 19547, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3171, 19712, F3, 28) (dual of [19712, 19541, 29]-code), using
(171−28, 171, 8525)-Net over F3 — Digital
Digital (143, 171, 8525)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3171, 8525, F3, 2, 28) (dual of [(8525, 2), 16879, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3171, 9859, F3, 2, 28) (dual of [(9859, 2), 19547, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3171, 19718, F3, 28) (dual of [19718, 19547, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3171, 19718, F3, 28) (dual of [19718, 19547, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(3171, 9859, F3, 2, 28) (dual of [(9859, 2), 19547, 29]-NRT-code), using
(171−28, 171, 2032919)-Net in Base 3 — Upper bound on s
There is no (143, 171, 2032920)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3870 223113 533716 527779 683265 657751 010950 153432 113032 978010 088973 041655 072358 656017 > 3171 [i]