Best Known (147, 147+41, s)-Nets in Base 4
(147, 147+41, 1052)-Net over F4 — Constructive and digital
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
(147, 147+41, 3924)-Net over F4 — Digital
Digital (147, 188, 3924)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4188, 3924, F4, 41) (dual of [3924, 3736, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, 4122, F4, 41) (dual of [4122, 3934, 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, 24, F4, 3) (dual of [24, 19, 4]-code or 24-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, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4188, 4122, F4, 41) (dual of [4122, 3934, 42]-code), using
(147, 147+41, 1178799)-Net in Base 4 — Upper bound on s
There is no (147, 188, 1178800)-net in base 4, because
- 1 times m-reduction [i] would yield (147, 187, 1178800)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 38478 855171 184529 828925 478577 834047 933605 782405 445940 680899 327997 773457 604947 298991 104363 429601 098224 860885 012266 > 4187 [i]