Best Known (225−37, 225, s)-Nets in Base 3
(225−37, 225, 1480)-Net over F3 — Constructive and digital
Digital (188, 225, 1480)-net over F3, using
- t-expansion [i] based on digital (187, 225, 1480)-net over F3, using
- 3 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 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, 57, 370)-net over F81, using
- 3 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
(225−37, 225, 9083)-Net over F3 — Digital
Digital (188, 225, 9083)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 9083, F3, 2, 37) (dual of [(9083, 2), 17941, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3225, 9859, F3, 2, 37) (dual of [(9859, 2), 19493, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3225, 19718, F3, 37) (dual of [19718, 19493, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [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(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(3225, 19718, F3, 37) (dual of [19718, 19493, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(3225, 9859, F3, 2, 37) (dual of [(9859, 2), 19493, 38]-NRT-code), using
(225−37, 225, 3270443)-Net in Base 3 — Upper bound on s
There is no (188, 225, 3270444)-net in base 3, because
- 1 times m-reduction [i] would yield (188, 224, 3270444)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 75017 336255 883902 777064 135917 752914 447634 899131 523217 589734 769181 570210 522267 890220 476145 931610 196864 296745 > 3224 [i]