Best Known (57, 57+18, s)-Nets in Base 4
(57, 57+18, 1028)-Net over F4 — Constructive and digital
Digital (57, 75, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (57, 76, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 257)-net over F256, using
(57, 57+18, 1155)-Net over F4 — Digital
Digital (57, 75, 1155)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(475, 1155, F4, 18) (dual of [1155, 1080, 19]-code), using
- 117 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 18 times 0, 1, 30 times 0, 1, 45 times 0) [i] based on linear OA(466, 1029, F4, 18) (dual of [1029, 963, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(466, 1024, F4, 18) (dual of [1024, 958, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(461, 1024, F4, 17) (dual of [1024, 963, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 117 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 18 times 0, 1, 30 times 0, 1, 45 times 0) [i] based on linear OA(466, 1029, F4, 18) (dual of [1029, 963, 19]-code), using
(57, 57+18, 143805)-Net in Base 4 — Upper bound on s
There is no (57, 75, 143806)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1427 287852 046309 385456 364592 145829 729424 167927 > 475 [i]