Best Known (84−25, 84, s)-Nets in Base 5
(84−25, 84, 264)-Net over F5 — Constructive and digital
Digital (59, 84, 264)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-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 (2, 14, 12)-net over F5, using
(84−25, 84, 699)-Net over F5 — Digital
Digital (59, 84, 699)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 699, F5, 25) (dual of [699, 615, 26]-code), using
- 71 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 7 times 0, 1, 22 times 0, 1, 37 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]
- 71 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 7 times 0, 1, 22 times 0, 1, 37 times 0) [i] based on linear OA(580, 624, F5, 25) (dual of [624, 544, 26]-code), using
(84−25, 84, 90323)-Net in Base 5 — Upper bound on s
There is no (59, 84, 90324)-net in base 5, because
- 1 times m-reduction [i] would yield (59, 83, 90324)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 10339 779363 829257 279406 863511 488556 972561 007441 880247 343873 > 583 [i]