Best Known (225−48, 225, s)-Nets in Base 3
(225−48, 225, 688)-Net over F3 — Constructive and digital
Digital (177, 225, 688)-net over F3, using
- 31 times duplication [i] based on digital (176, 224, 688)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (175, 224, 688)-net over F3, using
(225−48, 225, 1852)-Net over F3 — Digital
Digital (177, 225, 1852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3225, 1852, F3, 48) (dual of [1852, 1627, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 2194, F3, 48) (dual of [2194, 1969, 49]-code), using
- 1 times truncation [i] based on linear OA(3226, 2195, F3, 49) (dual of [2195, 1969, 50]-code), using
- construction X applied to Ce(48) ⊂ Ce(46) [i] based on
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- 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(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(48) ⊂ Ce(46) [i] based on
- 1 times truncation [i] based on linear OA(3226, 2195, F3, 49) (dual of [2195, 1969, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 2194, F3, 48) (dual of [2194, 1969, 49]-code), using
(225−48, 225, 145637)-Net in Base 3 — Upper bound on s
There is no (177, 225, 145638)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 225055 060562 550493 163399 303936 629883 988672 793573 226950 741820 961252 656752 621749 866355 217247 576610 735138 180721 > 3225 [i]