Best Known (146, 177, s)-Nets in Base 4
(146, 177, 1539)-Net over F4 — Constructive and digital
Digital (146, 177, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 59, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(146, 177, 16442)-Net over F4 — Digital
Digital (146, 177, 16442)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 16442, F4, 31) (dual of [16442, 16265, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4176, 16440, F4, 31) (dual of [16440, 16264, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4176, 16441, F4, 30) (dual of [16441, 16265, 31]-code), using Gilbert–Varšamov bound and bm = 4176 > Vbs−1(k−1) = 1382 466253 190924 163828 118526 786517 027729 241747 118198 040792 321081 103193 507061 702150 618821 241794 685777 368860 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4176, 16440, F4, 31) (dual of [16440, 16264, 32]-code), using
- construction X with Varšamov bound [i] based on
(146, 177, large)-Net in Base 4 — Upper bound on s
There is no (146, 177, large)-net in base 4, because
- 29 times m-reduction [i] would yield (146, 148, large)-net in base 4, but