Best Known (63, 63+31, s)-Nets in Base 16
(63, 63+31, 1030)-Net over F16 — Constructive and digital
Digital (63, 94, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (33, 64, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 32, 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, 32, 258)-net over F256, using
- digital (15, 30, 514)-net over F16, using
(63, 63+31, 4784)-Net over F16 — Digital
Digital (63, 94, 4784)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1694, 4784, F16, 31) (dual of [4784, 4690, 32]-code), using
- 679 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 20 times 0, 1, 76 times 0, 1, 210 times 0, 1, 364 times 0) [i] based on linear OA(1688, 4099, F16, 31) (dual of [4099, 4011, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(1688, 4096, F16, 31) (dual of [4096, 4008, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(1685, 4096, F16, 30) (dual of [4096, 4011, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- 679 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 20 times 0, 1, 76 times 0, 1, 210 times 0, 1, 364 times 0) [i] based on linear OA(1688, 4099, F16, 31) (dual of [4099, 4011, 32]-code), using
(63, 63+31, large)-Net in Base 16 — Upper bound on s
There is no (63, 94, large)-net in base 16, because
- 29 times m-reduction [i] would yield (63, 65, large)-net in base 16, but