Best Known (172, 172+44, s)-Nets in Base 3
(172, 172+44, 688)-Net over F3 — Constructive and digital
Digital (172, 216, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (172, 220, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 55, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 55, 172)-net over F81, using
(172, 172+44, 2247)-Net over F3 — Digital
Digital (172, 216, 2247)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 2247, F3, 44) (dual of [2247, 2031, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 2258, F3, 44) (dual of [2258, 2042, 45]-code), using
- 52 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0) [i] based on linear OA(3204, 2194, F3, 44) (dual of [2194, 1990, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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(43) ⊂ Ce(42) [i] based on
- 52 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0) [i] based on linear OA(3204, 2194, F3, 44) (dual of [2194, 1990, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 2258, F3, 44) (dual of [2258, 2042, 45]-code), using
(172, 172+44, 218898)-Net in Base 3 — Upper bound on s
There is no (172, 216, 218899)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 433909 852048 155753 276285 802679 771415 237938 344585 165884 592310 194651 823436 369796 319094 612774 481738 452637 > 3216 [i]