Best Known (178, 178+32, s)-Nets in Base 3
(178, 178+32, 1488)-Net over F3 — Constructive and digital
Digital (178, 210, 1488)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 18, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (160, 192, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
- digital (2, 18, 8)-net over F3, using
(178, 178+32, 12668)-Net over F3 — Digital
Digital (178, 210, 12668)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 12668, F3, 32) (dual of [12668, 12458, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 19748, F3, 32) (dual of [19748, 19538, 33]-code), using
- 5 times code embedding in larger space [i] based on linear OA(3205, 19743, F3, 32) (dual of [19743, 19538, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(315, 60, F3, 6) (dual of [60, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(3205, 19743, F3, 32) (dual of [19743, 19538, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 19748, F3, 32) (dual of [19748, 19538, 33]-code), using
(178, 178+32, 6219050)-Net in Base 3 — Upper bound on s
There is no (178, 210, 6219051)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 15684 242493 098385 395307 733541 771841 164785 254372 724793 907844 120838 869853 493111 822879 990513 050972 250081 > 3210 [i]