Best Known (85−25, 85, s)-Nets in Base 5
(85−25, 85, 268)-Net over F5 — Constructive and digital
Digital (60, 85, 268)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (45, 70, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 35, 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, 35, 126)-net over F25, using
- digital (3, 15, 16)-net over F5, using
(85−25, 85, 745)-Net over F5 — Digital
Digital (60, 85, 745)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(585, 745, F5, 25) (dual of [745, 660, 26]-code), using
- 116 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 7 times 0, 1, 22 times 0, 1, 37 times 0, 1, 44 times 0) [i] based on linear OA(580, 624, F5, 25) (dual of [624, 544, 26]-code), using
- 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]
- 116 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 7 times 0, 1, 22 times 0, 1, 37 times 0, 1, 44 times 0) [i] based on linear OA(580, 624, F5, 25) (dual of [624, 544, 26]-code), using
(85−25, 85, 103289)-Net in Base 5 — Upper bound on s
There is no (60, 85, 103290)-net in base 5, because
- 1 times m-reduction [i] would yield (60, 84, 103290)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 51702 447327 948800 250832 976046 633397 222220 565919 423527 719361 > 584 [i]