Best Known (159−30, 159, s)-Nets in Base 4
(159−30, 159, 1118)-Net over F4 — Constructive and digital
Digital (129, 159, 1118)-net over F4, using
- 41 times duplication [i] based on digital (128, 158, 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 (90, 120, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- digital (23, 38, 90)-net over F4, using
- (u, u+v)-construction [i] based on
(159−30, 159, 9380)-Net over F4 — Digital
Digital (129, 159, 9380)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 9380, F4, 30) (dual of [9380, 9221, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 16403, F4, 30) (dual of [16403, 16244, 31]-code), using
- construction XX applied to Ce(29) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(43, 18, F4, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(29) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4159, 16403, F4, 30) (dual of [16403, 16244, 31]-code), using
(159−30, 159, 5157987)-Net in Base 4 — Upper bound on s
There is no (129, 159, 5157988)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 533997 565015 824774 318942 791574 907466 416560 247829 761671 346252 215334 453748 234767 453199 003923 226200 > 4159 [i]