Best Known (221−37, 221, s)-Nets in Base 3
(221−37, 221, 1480)-Net over F3 — Constructive and digital
Digital (184, 221, 1480)-net over F3, using
- 3 times m-reduction [i] based on digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 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, 56, 370)-net over F81, using
(221−37, 221, 7978)-Net over F3 — Digital
Digital (184, 221, 7978)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 7978, F3, 2, 37) (dual of [(7978, 2), 15735, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 9852, F3, 2, 37) (dual of [(9852, 2), 19483, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3221, 19704, F3, 37) (dual of [19704, 19483, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 19705, F3, 37) (dual of [19705, 19484, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 19705, F3, 37) (dual of [19705, 19484, 38]-code), using
- OOA 2-folding [i] based on linear OA(3221, 19704, F3, 37) (dual of [19704, 19483, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 9852, F3, 2, 37) (dual of [(9852, 2), 19483, 38]-NRT-code), using
(221−37, 221, 2561999)-Net in Base 3 — Upper bound on s
There is no (184, 221, 2562000)-net in base 3, because
- 1 times m-reduction [i] would yield (184, 220, 2562000)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 139816 139718 342909 629662 579812 335207 805223 354944 648368 796512 654374 359080 890380 432403 672351 959748 415201 > 3220 [i]