Best Known (38, 38+15, s)-Nets in Base 5
(38, 38+15, 252)-Net over F5 — Constructive and digital
Digital (38, 53, 252)-net over F5, using
- 3 times m-reduction [i] based on digital (38, 56, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 28, 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, 28, 126)-net over F25, using
(38, 38+15, 706)-Net over F5 — Digital
Digital (38, 53, 706)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(553, 706, F5, 15) (dual of [706, 653, 16]-code), using
- 77 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 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]
- 77 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 22 times 0, 1, 40 times 0) [i] based on linear OA(548, 624, F5, 15) (dual of [624, 576, 16]-code), using
(38, 38+15, 131580)-Net in Base 5 — Upper bound on s
There is no (38, 53, 131581)-net in base 5, because
- 1 times m-reduction [i] would yield (38, 52, 131581)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2 220503 297724 070578 148248 089812 591325 > 552 [i]