Best Known (131−15, 131, s)-Nets in Base 8
(131−15, 131, 2396902)-Net over F8 — Constructive and digital
Digital (116, 131, 2396902)-net over F8, using
- 81 times duplication [i] based on digital (115, 130, 2396902)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (9, 16, 160)-net over F8, using
- (u, u+v)-construction [i] based on
(131−15, 131, large)-Net over F8 — Digital
Digital (116, 131, large)-net over F8, using
- t-expansion [i] based on digital (113, 131, large)-net over F8, using
- 1 times m-reduction [i] based on digital (113, 132, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8132, large, F8, 19) (dual of [large, large−132, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(8129, large, F8, 19) (dual of [large, large−129, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 3 times code embedding in larger space [i] based on linear OA(8129, large, F8, 19) (dual of [large, large−129, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8132, large, F8, 19) (dual of [large, large−132, 20]-code), using
- 1 times m-reduction [i] based on digital (113, 132, large)-net over F8, using
(131−15, 131, large)-Net in Base 8 — Upper bound on s
There is no (116, 131, large)-net in base 8, because
- 13 times m-reduction [i] would yield (116, 118, large)-net in base 8, but