Best Known (171, 171+34, s)-Nets in Base 3
(171, 171+34, 1480)-Net over F3 — Constructive and digital
Digital (171, 205, 1480)-net over F3, using
- 31 times duplication [i] based on digital (170, 204, 1480)-net over F3, using
- t-expansion [i] based on digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- t-expansion [i] based on digital (169, 204, 1480)-net over F3, using
(171, 171+34, 8238)-Net over F3 — Digital
Digital (171, 205, 8238)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 8238, F3, 2, 34) (dual of [(8238, 2), 16271, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 9853, F3, 2, 34) (dual of [(9853, 2), 19501, 35]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3204, 9853, F3, 2, 34) (dual of [(9853, 2), 19502, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3204, 19706, F3, 34) (dual of [19706, 19502, 35]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3203, 19705, F3, 34) (dual of [19705, 19502, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3203, 19705, F3, 34) (dual of [19705, 19502, 35]-code), using
- OOA 2-folding [i] based on linear OA(3204, 19706, F3, 34) (dual of [19706, 19502, 35]-code), using
- 31 times duplication [i] based on linear OOA(3204, 9853, F3, 2, 34) (dual of [(9853, 2), 19502, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 9853, F3, 2, 34) (dual of [(9853, 2), 19501, 35]-NRT-code), using
(171, 171+34, 2034382)-Net in Base 3 — Upper bound on s
There is no (171, 205, 2034383)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 64 544241 014714 632702 385321 860177 782026 147745 357245 919580 976026 226095 027475 687305 735441 887467 459967 > 3205 [i]