Best Known (86, 86+25, s)-Nets in Base 5
(86, 86+25, 416)-Net over F5 — Constructive and digital
Digital (86, 111, 416)-net over F5, using
- 51 times duplication [i] based on digital (85, 110, 416)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (30, 42, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 21, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 21, 104)-net over F25, using
- digital (43, 68, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 34, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- trace code for nets [i] based on digital (9, 34, 104)-net over F25, using
- digital (30, 42, 208)-net over F5, using
- (u, u+v)-construction [i] based on
(86, 86+25, 4201)-Net over F5 — Digital
Digital (86, 111, 4201)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5111, 4201, F5, 25) (dual of [4201, 4090, 26]-code), using
- 1066 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 52 times 0, 1, 97 times 0, 1, 154 times 0, 1, 205 times 0, 1, 241 times 0, 1, 266 times 0) [i] based on linear OA(5100, 3124, F5, 25) (dual of [3124, 3024, 26]-code), using
- 1 times truncation [i] based on linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 1 times truncation [i] based on linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using
- 1066 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 52 times 0, 1, 97 times 0, 1, 154 times 0, 1, 205 times 0, 1, 241 times 0, 1, 266 times 0) [i] based on linear OA(5100, 3124, F5, 25) (dual of [3124, 3024, 26]-code), using
(86, 86+25, 3376955)-Net in Base 5 — Upper bound on s
There is no (86, 111, 3376956)-net in base 5, because
- 1 times m-reduction [i] would yield (86, 110, 3376956)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 77037 268209 015564 721928 297917 881367 721309 584418 906197 146994 187032 010035 399425 > 5110 [i]