Best Known (112, 112+35, s)-Nets in Base 5
(112, 112+35, 504)-Net over F5 — Constructive and digital
Digital (112, 147, 504)-net over F5, using
- 1 times m-reduction [i] based on digital (112, 148, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (38, 56, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 28, 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, 28, 126)-net over F25, using
- digital (56, 92, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 46, 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, 46, 126)-net over F25, using
- digital (38, 56, 252)-net over F5, using
- (u, u+v)-construction [i] based on
(112, 112+35, 3580)-Net over F5 — Digital
Digital (112, 147, 3580)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5147, 3580, F5, 35) (dual of [3580, 3433, 36]-code), using
- 449 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 49 times 0, 1, 92 times 0, 1, 129 times 0, 1, 153 times 0) [i] based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- 449 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 49 times 0, 1, 92 times 0, 1, 129 times 0, 1, 153 times 0) [i] based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
(112, 112+35, 1806346)-Net in Base 5 — Upper bound on s
There is no (112, 147, 1806347)-net in base 5, because
- 1 times m-reduction [i] would yield (112, 146, 1806347)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 121041 015520 359939 556926 208251 238906 374179 851059 997013 156565 987125 100025 908763 503283 403054 847893 249005 > 5146 [i]