Best Known (226−38, 226, s)-Nets in Base 3
(226−38, 226, 1480)-Net over F3 — Constructive and digital
Digital (188, 226, 1480)-net over F3, using
- t-expansion [i] based on digital (187, 226, 1480)-net over F3, using
- 2 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
- 2 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
(226−38, 226, 7833)-Net over F3 — Digital
Digital (188, 226, 7833)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3226, 7833, F3, 2, 38) (dual of [(7833, 2), 15440, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3226, 9846, F3, 2, 38) (dual of [(9846, 2), 19466, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3226, 19692, F3, 38) (dual of [19692, 19466, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- OOA 2-folding [i] based on linear OA(3226, 19692, F3, 38) (dual of [19692, 19466, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(3226, 9846, F3, 2, 38) (dual of [(9846, 2), 19466, 39]-NRT-code), using
(226−38, 226, 1876778)-Net in Base 3 — Upper bound on s
There is no (188, 226, 1876779)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 675159 223676 727877 839212 535078 656797 143552 632218 369713 253421 691042 318100 716973 390188 281942 046255 571648 055211 > 3226 [i]