Best Known (212−34, 212, s)-Nets in Base 3
(212−34, 212, 1480)-Net over F3 — Constructive and digital
Digital (178, 212, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 54, 370)-net over F81, using
(212−34, 212, 9866)-Net over F3 — Digital
Digital (178, 212, 9866)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 9866, F3, 2, 34) (dual of [(9866, 2), 19520, 35]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3210, 9865, F3, 2, 34) (dual of [(9865, 2), 19520, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3210, 19730, F3, 34) (dual of [19730, 19520, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(311, 47, F3, 5) (dual of [47, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3210, 19730, F3, 34) (dual of [19730, 19520, 35]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3210, 9865, F3, 2, 34) (dual of [(9865, 2), 19520, 35]-NRT-code), using
(212−34, 212, 3198126)-Net in Base 3 — Upper bound on s
There is no (178, 212, 3198127)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 141158 326948 634425 890109 038066 214889 151562 664261 519226 855839 398899 098774 237271 413354 990200 416225 732927 > 3212 [i]