Best Known (39, 39+15, s)-Nets in Base 5
(39, 39+15, 252)-Net over F5 — Constructive and digital
Digital (39, 54, 252)-net over F5, using
- 4 times m-reduction [i] based on digital (39, 58, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 29, 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, 29, 126)-net over F25, using
(39, 39+15, 770)-Net over F5 — Digital
Digital (39, 54, 770)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(554, 770, F5, 15) (dual of [770, 716, 16]-code), using
- 140 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 times 0, 1, 62 times 0) [i] based on linear OA(548, 624, F5, 15) (dual of [624, 576, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 140 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 times 0, 1, 62 times 0) [i] based on linear OA(548, 624, F5, 15) (dual of [624, 576, 16]-code), using
(39, 39+15, 165595)-Net in Base 5 — Upper bound on s
There is no (39, 54, 165596)-net in base 5, because
- 1 times m-reduction [i] would yield (39, 53, 165596)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 11 102620 551358 525812 429643 497680 843505 > 553 [i]