Best Known (54, 54+26, s)-Nets in Base 16
(54, 54+26, 1030)-Net over F16 — Constructive and digital
Digital (54, 80, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 257)-net over F256, using
- digital (28, 54, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 27, 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, 27, 258)-net over F256, using
- digital (13, 26, 514)-net over F16, using
(54, 54+26, 4867)-Net over F16 — Digital
Digital (54, 80, 4867)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1680, 4867, F16, 26) (dual of [4867, 4787, 27]-code), using
- 761 step Varšamov–Edel lengthening with (ri) = (2, 1, 4 times 0, 1, 20 times 0, 1, 77 times 0, 1, 228 times 0, 1, 426 times 0) [i] based on linear OA(1673, 4099, F16, 26) (dual of [4099, 4026, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(1673, 4096, F16, 26) (dual of [4096, 4023, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1670, 4096, F16, 25) (dual of [4096, 4026, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- 761 step Varšamov–Edel lengthening with (ri) = (2, 1, 4 times 0, 1, 20 times 0, 1, 77 times 0, 1, 228 times 0, 1, 426 times 0) [i] based on linear OA(1673, 4099, F16, 26) (dual of [4099, 4026, 27]-code), using
(54, 54+26, large)-Net in Base 16 — Upper bound on s
There is no (54, 80, large)-net in base 16, because
- 24 times m-reduction [i] would yield (54, 56, large)-net in base 16, but