Best Known (135, 135+15, s)-Nets in Base 5
(135, 135+15, 2396752)-Net over F5 — Constructive and digital
Digital (135, 150, 2396752)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (127, 142, 2396742)-net over F5, using
- trace code for nets [i] based on digital (56, 71, 1198371)-net over F25, using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- trace code for nets [i] based on digital (56, 71, 1198371)-net over F25, using
- digital (1, 8, 10)-net over F5, using
(135, 135+15, large)-Net over F5 — Digital
Digital (135, 150, large)-net over F5, using
- 54 times duplication [i] based on digital (131, 146, large)-net over F5, using
- t-expansion [i] based on digital (129, 146, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5146, large, F5, 17) (dual of [large, large−146, 18]-code), using
- 15 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 15 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5146, large, F5, 17) (dual of [large, large−146, 18]-code), using
- t-expansion [i] based on digital (129, 146, large)-net over F5, using
(135, 135+15, large)-Net in Base 5 — Upper bound on s
There is no (135, 150, large)-net in base 5, because
- 13 times m-reduction [i] would yield (135, 137, large)-net in base 5, but