Best Known (211−46, 211, s)-Nets in Base 3
(211−46, 211, 640)-Net over F3 — Constructive and digital
Digital (165, 211, 640)-net over F3, using
- t-expansion [i] based on digital (164, 211, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (164, 212, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 53, 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, 53, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (164, 212, 640)-net over F3, using
(211−46, 211, 1593)-Net over F3 — Digital
Digital (165, 211, 1593)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3211, 1593, F3, 46) (dual of [1593, 1382, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- discarding factors / shortening the dual code based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
(211−46, 211, 112307)-Net in Base 3 — Upper bound on s
There is no (165, 211, 112308)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 47058 948634 499823 651264 861139 313000 526857 413909 945586 857417 883584 382451 304435 757404 292004 957855 963057 > 3211 [i]