Best Known (113, 136, s)-Nets in Base 4
(113, 136, 2056)-Net over F4 — Constructive and digital
Digital (113, 136, 2056)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (33, 44, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
- digital (69, 92, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 23, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 23, 257)-net over F256, using
- digital (33, 44, 1028)-net over F4, using
(113, 136, 16444)-Net over F4 — Digital
Digital (113, 136, 16444)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4136, 16444, F4, 23) (dual of [16444, 16308, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4134, 16440, F4, 23) (dual of [16440, 16306, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(14) [i] based on
- 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(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(22) ⊂ Ce(14) [i] based on
- linear OA(4134, 16442, F4, 22) (dual of [16442, 16308, 23]-code), using Gilbert–Varšamov bound and bm = 4134 > Vbs−1(k−1) = 6 923582 874662 785998 435914 915822 848927 272983 571982 036525 921694 273644 644840 923356 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(4134, 16440, F4, 23) (dual of [16440, 16306, 24]-code), using
- construction X with Varšamov bound [i] based on
(113, 136, large)-Net in Base 4 — Upper bound on s
There is no (113, 136, large)-net in base 4, because
- 21 times m-reduction [i] would yield (113, 115, large)-net in base 4, but