Best Known (39−8, 39, s)-Nets in Base 4
(39−8, 39, 1071)-Net over F4 — Constructive and digital
Digital (31, 39, 1071)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 43)-net over F4, using
- digital (24, 32, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
(39−8, 39, 4126)-Net over F4 — Digital
Digital (31, 39, 4126)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(439, 4126, F4, 8) (dual of [4126, 4087, 9]-code), using
- 15 step Varšamov–Edel lengthening with (ri) = (1, 14 times 0) [i] based on linear OA(438, 4110, F4, 8) (dual of [4110, 4072, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(437, 4096, F4, 9) (dual of [4096, 4059, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- 15 step Varšamov–Edel lengthening with (ri) = (1, 14 times 0) [i] based on linear OA(438, 4110, F4, 8) (dual of [4110, 4072, 9]-code), using
(39−8, 39, 547033)-Net in Base 4 — Upper bound on s
There is no (31, 39, 547034)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 302231 728456 995460 142554 > 439 [i]