Best Known (180, 180+46, s)-Nets in Base 3
(180, 180+46, 688)-Net over F3 — Constructive and digital
Digital (180, 226, 688)-net over F3, using
- t-expansion [i] based on digital (178, 226, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 57, 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, 57, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
(180, 180+46, 2299)-Net over F3 — Digital
Digital (180, 226, 2299)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 2299, F3, 46) (dual of [2299, 2073, 47]-code), using
- 97 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0) [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]
- 97 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 23 times 0) [i] based on linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using
(180, 180+46, 229942)-Net in Base 3 — Upper bound on s
There is no (180, 226, 229943)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 675207 459932 214652 257421 949154 568315 605444 233689 953317 485995 628466 514790 370235 631330 964705 722745 778659 542323 > 3226 [i]