Best Known (133−31, 133, s)-Nets in Base 5
(133−31, 133, 504)-Net over F5 — Constructive and digital
Digital (102, 133, 504)-net over F5, using
- 51 times duplication [i] based on digital (101, 132, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (35, 50, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 25, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 25, 126)-net over F25, using
- digital (51, 82, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 41, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- trace code for nets [i] based on digital (10, 41, 126)-net over F25, using
- digital (35, 50, 252)-net over F5, using
- (u, u+v)-construction [i] based on
(133−31, 133, 3803)-Net over F5 — Digital
Digital (102, 133, 3803)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5133, 3803, F5, 31) (dual of [3803, 3670, 32]-code), using
- 665 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0, 1, 32 times 0, 1, 54 times 0, 1, 85 times 0, 1, 120 times 0, 1, 155 times 0, 1, 181 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 665 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0, 1, 32 times 0, 1, 54 times 0, 1, 85 times 0, 1, 120 times 0, 1, 155 times 0, 1, 181 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
(133−31, 133, 2273215)-Net in Base 5 — Upper bound on s
There is no (102, 133, 2273216)-net in base 5, because
- 1 times m-reduction [i] would yield (102, 132, 2273216)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 183 671438 610791 798718 474867 002136 806802 345956 439912 706970 596156 325337 652730 193372 189963 403009 > 5132 [i]