Best Known (42, 58, s)-Nets in Base 5
(42, 58, 252)-Net over F5 — Constructive and digital
Digital (42, 58, 252)-net over F5, using
- 6 times m-reduction [i] based on digital (42, 64, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 32, 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, 32, 126)-net over F25, using
(42, 58, 824)-Net over F5 — Digital
Digital (42, 58, 824)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(558, 824, F5, 16) (dual of [824, 766, 17]-code), using
- 190 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 33 times 0, 1, 51 times 0, 1, 70 times 0) [i] based on linear OA(549, 625, F5, 16) (dual of [625, 576, 17]-code), using
- an extension Ce(15) of 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]
- 190 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 33 times 0, 1, 51 times 0, 1, 70 times 0) [i] based on linear OA(549, 625, F5, 16) (dual of [625, 576, 17]-code), using
(42, 58, 109936)-Net in Base 5 — Upper bound on s
There is no (42, 58, 109937)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 34696 388321 297771 668593 765287 623537 168865 > 558 [i]