Best Known (178, 178+48, s)-Nets in Base 3
(178, 178+48, 688)-Net over F3 — Constructive and digital
Digital (178, 226, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (178, 228, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 57, 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, 57, 172)-net over F81, using
(178, 178+48, 1898)-Net over F3 — Digital
Digital (178, 226, 1898)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 1898, F3, 48) (dual of [1898, 1672, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 2202, F3, 48) (dual of [2202, 1976, 49]-code), using
- construction X applied to Ce(48) ⊂ Ce(45) [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(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 2202, F3, 48) (dual of [2202, 1976, 49]-code), using
(178, 178+48, 152460)-Net in Base 3 — Upper bound on s
There is no (178, 226, 152461)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 675197 651503 518708 834388 294909 099287 329961 475696 005461 646487 409319 580635 204176 743550 669731 286456 619781 814049 > 3226 [i]