Best Known (53, 78, s)-Nets in Base 16
(53, 78, 1032)-Net over F16 — Constructive and digital
Digital (53, 78, 1032)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (14, 26, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 13, 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, 13, 258)-net over F256, using
- digital (27, 52, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 26, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- trace code for nets [i] based on digital (1, 26, 258)-net over F256, using
- digital (14, 26, 516)-net over F16, using
(53, 78, 5369)-Net over F16 — Digital
Digital (53, 78, 5369)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1678, 5369, F16, 25) (dual of [5369, 5291, 26]-code), using
- 1262 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 16 times 0, 1, 64 times 0, 1, 200 times 0, 1, 411 times 0, 1, 562 times 0) [i] based on linear OA(1670, 4099, F16, 25) (dual of [4099, 4029, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- 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(1667, 4096, F16, 24) (dual of [4096, 4029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- 1262 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 16 times 0, 1, 64 times 0, 1, 200 times 0, 1, 411 times 0, 1, 562 times 0) [i] based on linear OA(1670, 4099, F16, 25) (dual of [4099, 4029, 26]-code), using
(53, 78, large)-Net in Base 16 — Upper bound on s
There is no (53, 78, large)-net in base 16, because
- 23 times m-reduction [i] would yield (53, 55, large)-net in base 16, but