Best Known (163−31, 163, s)-Nets in Base 4
(163−31, 163, 1118)-Net over F4 — Constructive and digital
Digital (132, 163, 1118)-net over F4, using
- 41 times duplication [i] based on digital (131, 162, 1118)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (23, 38, 90)-net over F4, using
- trace code for nets [i] based on digital (4, 19, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 19, 45)-net over F16, using
- digital (93, 124, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- digital (23, 38, 90)-net over F4, using
- (u, u+v)-construction [i] based on
(163−31, 163, 8956)-Net over F4 — Digital
Digital (132, 163, 8956)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4163, 8956, F4, 31) (dual of [8956, 8793, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 16399, F4, 31) (dual of [16399, 16236, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [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(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4163, 16399, F4, 31) (dual of [16399, 16236, 32]-code), using
(163−31, 163, 6806009)-Net in Base 4 — Upper bound on s
There is no (132, 163, 6806010)-net in base 4, because
- 1 times m-reduction [i] would yield (132, 162, 6806010)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 34 175858 942009 154366 344232 960285 802889 250307 373975 533363 140099 698393 923762 558267 786986 581631 025528 > 4162 [i]