Best Known (71, 71+31, s)-Nets in Base 5
(71, 71+31, 296)-Net over F5 — Constructive and digital
Digital (71, 102, 296)-net over F5, using
- 2 times m-reduction [i] based on digital (71, 104, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 52, 148)-net over F25, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 19 and N(F) ≥ 148, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- trace code for nets [i] based on digital (19, 52, 148)-net over F25, using
(71, 71+31, 733)-Net over F5 — Digital
Digital (71, 102, 733)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5102, 733, F5, 31) (dual of [733, 631, 32]-code), using
- 101 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 27 times 0, 1, 33 times 0) [i] based on linear OA(595, 625, F5, 31) (dual of [625, 530, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 101 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 27 times 0, 1, 33 times 0) [i] based on linear OA(595, 625, F5, 31) (dual of [625, 530, 32]-code), using
(71, 71+31, 81667)-Net in Base 5 — Upper bound on s
There is no (71, 102, 81668)-net in base 5, because
- 1 times m-reduction [i] would yield (71, 101, 81668)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 39449 041802 102324 404496 800042 896171 296001 752025 565300 553711 572197 925009 > 5101 [i]