Best Known (148, 189, s)-Nets in Base 4
(148, 189, 1052)-Net over F4 — Constructive and digital
Digital (148, 189, 1052)-net over F4, using
- 41 times duplication [i] based on digital (147, 188, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 47, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 47, 263)-net over F256, using
(148, 189, 4067)-Net over F4 — Digital
Digital (148, 189, 4067)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4189, 4067, F4, 41) (dual of [4067, 3878, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 4124, F4, 41) (dual of [4124, 3935, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(45, 25, F4, 3) (dual of [25, 20, 4]-code or 25-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 4124, F4, 41) (dual of [4124, 3935, 42]-code), using
(148, 189, 1263407)-Net in Base 4 — Upper bound on s
There is no (148, 189, 1263408)-net in base 4, because
- 1 times m-reduction [i] would yield (148, 188, 1263408)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 153916 107862 719963 620386 053394 824616 026795 482531 685409 256405 441455 227064 757466 255512 988755 142688 614626 196793 687474 > 4188 [i]