Best Known (42, 42+19, s)-Nets in Base 5
(42, 42+19, 252)-Net over F5 — Constructive and digital
Digital (42, 61, 252)-net over F5, using
- 3 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, 42+19, 515)-Net over F5 — Digital
Digital (42, 61, 515)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 515, F5, 19) (dual of [515, 454, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(561, 624, F5, 19) (dual of [624, 563, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(561, 624, F5, 19) (dual of [624, 563, 20]-code), using
(42, 42+19, 47362)-Net in Base 5 — Upper bound on s
There is no (42, 61, 47363)-net in base 5, because
- 1 times m-reduction [i] would yield (42, 60, 47363)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 867451 976356 433718 294299 911807 428869 389805 > 560 [i]