Best Known (70, 70+31, s)-Nets in Base 5
(70, 70+31, 296)-Net over F5 — Constructive and digital
Digital (70, 101, 296)-net over F5, using
- 1 times m-reduction [i] based on digital (70, 102, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 51, 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, 51, 148)-net over F25, using
(70, 70+31, 698)-Net over F5 — Digital
Digital (70, 101, 698)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5101, 698, F5, 31) (dual of [698, 597, 32]-code), using
- 67 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 27 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]
- 67 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 27 times 0) [i] based on linear OA(595, 625, F5, 31) (dual of [625, 530, 32]-code), using
(70, 70+31, 73357)-Net in Base 5 — Upper bound on s
There is no (70, 101, 73358)-net in base 5, because
- 1 times m-reduction [i] would yield (70, 100, 73358)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 7889 818401 739623 398239 380435 409961 090261 987227 719835 046580 655056 621145 > 5100 [i]