Best Known (192−41, 192, s)-Nets in Base 3
(192−41, 192, 688)-Net over F3 — Constructive and digital
Digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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
(192−41, 192, 1635)-Net over F3 — Digital
Digital (151, 192, 1635)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3192, 1635, F3, 41) (dual of [1635, 1443, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 2197, F3, 41) (dual of [2197, 2005, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(40) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3192, 2197, F3, 41) (dual of [2197, 2005, 42]-code), using
(192−41, 192, 149529)-Net in Base 3 — Upper bound on s
There is no (151, 192, 149530)-net in base 3, because
- 1 times m-reduction [i] would yield (151, 191, 149530)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13 495051 362139 205949 419205 869223 557560 312293 226486 578380 841458 421678 406296 920255 228379 485001 > 3191 [i]