Best Known (39, 59, s)-Nets in Base 16
(39, 59, 771)-Net over F16 — Constructive and digital
Digital (39, 59, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) for nets [i] based on digital (0, 10, 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
- base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) for nets [i] based on digital (0, 10, 257)-net over F256, using
- digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
- digital (9, 19, 257)-net over F16, using
(39, 59, 3810)-Net over F16 — Digital
Digital (39, 59, 3810)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1659, 3810, F16, 20) (dual of [3810, 3751, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1659, 4112, F16, 20) (dual of [4112, 4053, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(1655, 4096, F16, 20) (dual of [4096, 4041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1643, 4096, F16, 15) (dual of [4096, 4053, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(164, 16, F16, 4) (dual of [16, 12, 5]-code or 16-arc in PG(3,16)), using
- Reed–Solomon code RS(12,16) [i]
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(1659, 4112, F16, 20) (dual of [4112, 4053, 21]-code), using
(39, 59, 3838772)-Net in Base 16 — Upper bound on s
There is no (39, 59, 3838773)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 110428 130101 916509 150153 426328 155835 369309 455494 123523 045607 109277 626076 > 1659 [i]