Best Known (124, 158, s)-Nets in Base 3
(124, 158, 640)-Net over F3 — Constructive and digital
Digital (124, 158, 640)-net over F3, using
- 32 times duplication [i] based on digital (122, 156, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 39, 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, 39, 160)-net over F81, using
(124, 158, 1373)-Net over F3 — Digital
Digital (124, 158, 1373)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3158, 1373, F3, 34) (dual of [1373, 1215, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3158, 2200, F3, 34) (dual of [2200, 2042, 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3158, 2200, F3, 34) (dual of [2200, 2042, 35]-code), using
(124, 158, 97557)-Net in Base 3 — Upper bound on s
There is no (124, 158, 97558)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2427 498718 859240 915670 551155 196722 063339 333294 115474 444562 149954 591594 153997 > 3158 [i]