Best Known (31, 45, s)-Nets in Base 5
(31, 45, 144)-Net over F5 — Constructive and digital
Digital (31, 45, 144)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (22, 36, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F25, using
- digital (2, 9, 12)-net over F5, using
(31, 45, 476)-Net over F5 — Digital
Digital (31, 45, 476)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(545, 476, F5, 14) (dual of [476, 431, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(545, 624, F5, 14) (dual of [624, 579, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(545, 624, F5, 14) (dual of [624, 579, 15]-code), using
(31, 45, 26312)-Net in Base 5 — Upper bound on s
There is no (31, 45, 26313)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 28 426976 146991 797518 206627 907533 > 545 [i]