Best Known (80−27, 80, s)-Nets in Base 16
(80−27, 80, 1028)-Net over F16 — Constructive and digital
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
(80−27, 80, 4138)-Net over F16 — Digital
Digital (53, 80, 4138)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1680, 4138, F16, 27) (dual of [4138, 4058, 28]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0) [i] based on linear OA(1677, 4103, F16, 27) (dual of [4103, 4026, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(1676, 4096, F16, 27) (dual of [4096, 4020, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- 32 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0) [i] based on linear OA(1677, 4103, F16, 27) (dual of [4103, 4026, 28]-code), using
(80−27, 80, 7846194)-Net in Base 16 — Upper bound on s
There is no (53, 80, 7846195)-net in base 16, because
- 1 times m-reduction [i] would yield (53, 79, 7846195)-net in base 16, but
- 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]