Best Known (147, 147+31, s)-Nets in Base 3
(147, 147+31, 896)-Net over F3 — Constructive and digital
Digital (147, 178, 896)-net over F3, using
- 32 times duplication [i] based on digital (145, 176, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 44, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 44, 224)-net over F81, using
(147, 147+31, 4739)-Net over F3 — Digital
Digital (147, 178, 4739)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3178, 4739, F3, 31) (dual of [4739, 4561, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 6618, F3, 31) (dual of [6618, 6440, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(317, 57, F3, 7) (dual of [57, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3178, 6618, F3, 31) (dual of [6618, 6440, 32]-code), using
(147, 147+31, 1370133)-Net in Base 3 — Upper bound on s
There is no (147, 178, 1370134)-net in base 3, because
- 1 times m-reduction [i] would yield (147, 177, 1370134)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 821401 659142 884573 694374 012686 801144 642473 065779 547668 753408 827588 307518 807792 254601 > 3177 [i]