Best Known (147, 147+42, s)-Nets in Base 4
(147, 147+42, 1048)-Net over F4 — Constructive and digital
Digital (147, 189, 1048)-net over F4, using
- 41 times duplication [i] based on digital (146, 188, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 47, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 47, 262)-net over F256, using
(147, 147+42, 3520)-Net over F4 — Digital
Digital (147, 189, 3520)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4189, 3520, F4, 42) (dual of [3520, 3331, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 4105, F4, 42) (dual of [4105, 3916, 43]-code), using
- construction XX applied to Ce(41) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4187, 4096, F4, 42) (dual of [4096, 3909, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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(4175, 4096, F4, 39) (dual of [4096, 3921, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, 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(41) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 4105, F4, 42) (dual of [4105, 3916, 43]-code), using
(147, 147+42, 758405)-Net in Base 4 — Upper bound on s
There is no (147, 189, 758406)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 615664 100216 284727 426686 962274 651779 500523 394636 972187 751090 360644 390243 627916 035725 430128 297027 721330 350778 889736 > 4189 [i]