Best Known (192−40, 192, s)-Nets in Base 3
(192−40, 192, 688)-Net over F3 — Constructive and digital
Digital (152, 192, 688)-net over F3, using
- t-expansion [i] based on 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
- trace code for nets [i] based on digital (7, 48, 172)-net over F81, using
(192−40, 192, 1844)-Net over F3 — Digital
Digital (152, 192, 1844)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3192, 1844, F3, 40) (dual of [1844, 1652, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 2218, F3, 40) (dual of [2218, 2026, 41]-code), using
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- 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(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(39) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3192, 2218, F3, 40) (dual of [2218, 2026, 41]-code), using
(192−40, 192, 157974)-Net in Base 3 — Upper bound on s
There is no (152, 192, 157975)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 40 486908 665464 309812 549532 037902 441090 325240 245929 269238 801896 959953 130531 644027 886281 532761 > 3192 [i]