Best Known (124, 164, s)-Nets in Base 3
(124, 164, 400)-Net over F3 — Constructive and digital
Digital (124, 164, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 41, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(124, 164, 803)-Net over F3 — Digital
Digital (124, 164, 803)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3164, 803, F3, 40) (dual of [803, 639, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3164, 804, F3, 40) (dual of [804, 640, 41]-code), using
- 65 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 17 times 0) [i] based on linear OA(3154, 729, F3, 40) (dual of [729, 575, 41]-code), using
- an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 65 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 17 times 0) [i] based on linear OA(3154, 729, F3, 40) (dual of [729, 575, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3164, 804, F3, 40) (dual of [804, 640, 41]-code), using
(124, 164, 33917)-Net in Base 3 — Upper bound on s
There is no (124, 164, 33918)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 770398 509992 701018 650020 575756 201130 020884 507449 285324 197662 219267 908845 832873 > 3164 [i]