Best Known (109, 109+35, s)-Nets in Base 5
(109, 109+35, 504)-Net over F5 — Constructive and digital
Digital (109, 144, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (37, 54, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 27, 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, 27, 126)-net over F25, using
- digital (55, 90, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 45, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- trace code for nets [i] based on digital (10, 45, 126)-net over F25, using
- digital (37, 54, 252)-net over F5, using
(109, 109+35, 3200)-Net over F5 — Digital
Digital (109, 144, 3200)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5144, 3200, F5, 35) (dual of [3200, 3056, 36]-code), using
- 72 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 49 times 0) [i] based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 1 times truncation [i] based on linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using
- 72 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 49 times 0) [i] based on linear OA(5140, 3124, F5, 35) (dual of [3124, 2984, 36]-code), using
(109, 109+35, 1359729)-Net in Base 5 — Upper bound on s
There is no (109, 144, 1359730)-net in base 5, because
- 1 times m-reduction [i] would yield (109, 143, 1359730)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 8968 367668 462040 783419 848262 666031 744033 666277 698192 455655 805271 894600 810530 444649 530679 033041 802185 > 5143 [i]