Best Known (77−24, 77, s)-Nets in Base 5
(77−24, 77, 252)-Net over F5 — Constructive and digital
Digital (53, 77, 252)-net over F5, using
- 9 times m-reduction [i] based on digital (53, 86, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 43, 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, 43, 126)-net over F25, using
(77−24, 77, 574)-Net over F5 — Digital
Digital (53, 77, 574)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 574, F5, 24) (dual of [574, 497, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 624, F5, 24) (dual of [624, 547, 25]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(577, 624, F5, 24) (dual of [624, 547, 25]-code), using
(77−24, 77, 40389)-Net in Base 5 — Upper bound on s
There is no (53, 77, 40390)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 661859 378509 673078 398791 081554 301605 678712 813628 731841 > 577 [i]