Best Known (178, 178+49, s)-Nets in Base 3
(178, 178+49, 688)-Net over F3 — Constructive and digital
Digital (178, 227, 688)-net over F3, using
- 1 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+49, 1765)-Net over F3 — Digital
Digital (178, 227, 1765)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3227, 1765, F3, 49) (dual of [1765, 1538, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3227, 2197, F3, 49) (dual of [2197, 1970, 50]-code), using
- construction XX applied to Ce(48) ⊂ Ce(46) ⊂ 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(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [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, 9, F3, 1) (dual of [9, 8, 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
- 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(48) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(3227, 2197, F3, 49) (dual of [2197, 1970, 50]-code), using
(178, 178+49, 152460)-Net in Base 3 — Upper bound on s
There is no (178, 227, 152461)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 226, 152461)-net in base 3, but
- 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]