Best Known (181, 181+45, s)-Nets in Base 3
(181, 181+45, 688)-Net over F3 — Constructive and digital
Digital (181, 226, 688)-net over F3, using
- 6 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F81, using
(181, 181+45, 2464)-Net over F3 — Digital
Digital (181, 226, 2464)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 2464, F3, 45) (dual of [2464, 2238, 46]-code), using
- 262 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 25 times 0, 1, 32 times 0, 1, 39 times 0, 1, 47 times 0, 1, 52 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
- 1 times truncation [i] 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]
- 1 times truncation [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
- 262 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 25 times 0, 1, 32 times 0, 1, 39 times 0, 1, 47 times 0, 1, 52 times 0) [i] based on linear OA(3210, 2186, F3, 45) (dual of [2186, 1976, 46]-code), using
(181, 181+45, 343119)-Net in Base 3 — Upper bound on s
There is no (181, 226, 343120)-net in base 3, because
- 1 times m-reduction [i] would yield (181, 225, 343120)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 225063 306634 989529 542513 568478 908046 796090 487397 189004 275950 462130 839763 035109 819023 397515 323836 142369 508833 > 3225 [i]