Best Known (233−50, 233, s)-Nets in Base 3
(233−50, 233, 688)-Net over F3 — Constructive and digital
Digital (183, 233, 688)-net over F3, using
- 31 times duplication [i] based on digital (182, 232, 688)-net over F3, using
- t-expansion [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
- t-expansion [i] based on digital (181, 232, 688)-net over F3, using
(233−50, 233, 1852)-Net over F3 — Digital
Digital (183, 233, 1852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3233, 1852, F3, 50) (dual of [1852, 1619, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 2195, F3, 50) (dual of [2195, 1962, 51]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- 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(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(49) ⊂ Ce(48) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 2195, F3, 50) (dual of [2195, 1962, 51]-code), using
(233−50, 233, 142328)-Net in Base 3 — Upper bound on s
There is no (183, 233, 142329)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1476 752841 371275 779001 139390 824208 001612 427183 802062 134729 925292 720157 777743 319816 497951 700069 494300 864147 110419 > 3233 [i]