Best Known (53, 79, s)-Nets in Base 16
(53, 79, 1028)-Net over F16 — Constructive and digital
Digital (53, 79, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (53, 80, 1028)-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 (27, 54, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- digital (13, 26, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(53, 79, 4439)-Net over F16 — Digital
Digital (53, 79, 4439)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1679, 4439, F16, 26) (dual of [4439, 4360, 27]-code), using
- 334 step Varšamov–Edel lengthening with (ri) = (2, 1, 4 times 0, 1, 20 times 0, 1, 77 times 0, 1, 228 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
- 334 step Varšamov–Edel lengthening with (ri) = (2, 1, 4 times 0, 1, 20 times 0, 1, 77 times 0, 1, 228 times 0) [i] based on linear OA(1673, 4099, F16, 26) (dual of [4099, 4026, 27]-code), using
(53, 79, 7846194)-Net in Base 16 — Upper bound on s
There is no (53, 79, 7846195)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 133499 382871 301466 468669 949930 610512 893533 081032 495531 138360 715860 211948 972887 397965 591745 654776 > 1679 [i]