Best Known (42, 42+20, s)-Nets in Base 16
(42, 42+20, 1030)-Net over F16 — Constructive and digital
Digital (42, 62, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (10, 20, 514)-net over F16, using
- trace code 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
- trace code for nets [i] based on digital (0, 10, 257)-net over F256, using
- digital (22, 42, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F256, using
- digital (10, 20, 514)-net over F16, using
(42, 42+20, 4674)-Net over F16 — Digital
Digital (42, 62, 4674)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1662, 4674, F16, 20) (dual of [4674, 4612, 21]-code), using
- 568 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 154 times 0, 1, 337 times 0) [i] based on linear OA(1655, 4099, F16, 20) (dual of [4099, 4044, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [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(1652, 4096, F16, 19) (dual of [4096, 4044, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 568 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 154 times 0, 1, 337 times 0) [i] based on linear OA(1655, 4099, F16, 20) (dual of [4099, 4044, 21]-code), using
(42, 42+20, large)-Net in Base 16 — Upper bound on s
There is no (42, 62, large)-net in base 16, because
- 18 times m-reduction [i] would yield (42, 44, large)-net in base 16, but