Best Known (57−12, 57, s)-Nets in Base 4
(57−12, 57, 1046)-Net over F4 — Constructive and digital
Digital (45, 57, 1046)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 18)-net over F4, using
- digital (36, 48, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
(57−12, 57, 3544)-Net over F4 — Digital
Digital (45, 57, 3544)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(457, 3544, F4, 12) (dual of [3544, 3487, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(457, 4111, F4, 12) (dual of [4111, 4054, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(456, 4110, F4, 12) (dual of [4110, 4054, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(443, 4096, F4, 10) (dual of [4096, 4053, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(413, 14, F4, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,4)), using
- dual of repetition code with length 14 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 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 X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(456, 4110, F4, 12) (dual of [4110, 4054, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(457, 4111, F4, 12) (dual of [4111, 4054, 13]-code), using
(57−12, 57, 523199)-Net in Base 4 — Upper bound on s
There is no (45, 57, 523200)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 20769 395459 780377 673984 936116 397521 > 457 [i]