Best Known (254−46, 254, s)-Nets in Base 4
(254−46, 254, 1561)-Net over F4 — Constructive and digital
Digital (208, 254, 1561)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 32, 22)-net over F4, using
- net from sequence [i] based on digital (9, 21)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 9 and N(F) ≥ 22, using
- net from sequence [i] based on digital (9, 21)-sequence over F4, using
- digital (176, 222, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 74, 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
- trace code for nets [i] based on digital (28, 74, 513)-net over F64, using
- digital (9, 32, 22)-net over F4, using
(254−46, 254, 16442)-Net over F4 — Digital
Digital (208, 254, 16442)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4254, 16442, F4, 46) (dual of [16442, 16188, 47]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4253, 16440, F4, 46) (dual of [16440, 16187, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(37) [i] based on
- linear OA(4239, 16384, F4, 46) (dual of [16384, 16145, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(4197, 16384, F4, 38) (dual of [16384, 16187, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [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(45) ⊂ Ce(37) [i] based on
- linear OA(4253, 16441, F4, 45) (dual of [16441, 16188, 46]-code), using Gilbert–Varšamov bound and bm = 4253 > Vbs−1(k−1) = 110 609333 782011 556837 462355 978892 871974 061775 726667 881821 225847 938068 072305 225392 178123 242930 701054 057013 957173 110224 821398 319778 035081 229098 500844 144538 [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(4253, 16440, F4, 46) (dual of [16440, 16187, 47]-code), using
- construction X with Varšamov bound [i] based on
(254−46, 254, large)-Net in Base 4 — Upper bound on s
There is no (208, 254, large)-net in base 4, because
- 44 times m-reduction [i] would yield (208, 210, large)-net in base 4, but