Best Known (125, 125+34, s)-Nets in Base 3
(125, 125+34, 640)-Net over F3 — Constructive and digital
Digital (125, 159, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (125, 160, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 40, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 40, 160)-net over F81, using
(125, 125+34, 1422)-Net over F3 — Digital
Digital (125, 159, 1422)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 1422, F3, 34) (dual of [1422, 1263, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 2205, F3, 34) (dual of [2205, 2046, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(34, 18, F3, 2) (dual of [18, 14, 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
- discarding factors / shortening the dual code based on linear OA(3159, 2205, F3, 34) (dual of [2205, 2046, 35]-code), using
(125, 125+34, 104071)-Net in Base 3 — Upper bound on s
There is no (125, 159, 104072)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7282 627289 812898 821447 270090 897722 142576 996384 351110 054331 246424 768029 433361 > 3159 [i]