Best Known (85−26, 85, s)-Nets in Base 5
(85−26, 85, 258)-Net over F5 — Constructive and digital
Digital (59, 85, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (46, 72, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 36, 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, 36, 126)-net over F25, using
- digital (0, 13, 6)-net over F5, using
(85−26, 85, 650)-Net over F5 — Digital
Digital (59, 85, 650)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(585, 650, F5, 26) (dual of [650, 565, 27]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(581, 625, F5, 26) (dual of [625, 544, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0) [i] based on linear OA(581, 625, F5, 26) (dual of [625, 544, 27]-code), using
(85−26, 85, 52657)-Net in Base 5 — Upper bound on s
There is no (59, 85, 52658)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 258540 067960 993884 013239 645555 074401 153124 759704 316874 983625 > 585 [i]