Best Known (175, 222, s)-Nets in Base 3
(175, 222, 688)-Net over F3 — Constructive and digital
Digital (175, 222, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (175, 224, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 56, 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, 56, 172)-net over F81, using
(175, 222, 1901)-Net over F3 — Digital
Digital (175, 222, 1901)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3222, 1901, F3, 47) (dual of [1901, 1679, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 2205, F3, 47) (dual of [2205, 1983, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(43) [i] based on
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- 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(34, 18, F3, 2) (dual of [18, 14, 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(46) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 2205, F3, 47) (dual of [2205, 1983, 48]-code), using
(175, 222, 181086)-Net in Base 3 — Upper bound on s
There is no (175, 222, 181087)-net in base 3, because
- 1 times m-reduction [i] would yield (175, 221, 181087)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2778 713262 697418 456459 770238 818698 316622 941027 241655 992994 351052 511705 337797 885725 442194 608799 660109 806419 > 3221 [i]