Best Known (51, 77, s)-Nets in Base 16
(51, 77, 771)-Net over F16 — Constructive and digital
Digital (51, 77, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 25, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(12,256) in PG(24,16)) 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
- base reduction for projective spaces (embedding PG(12,256) in PG(24,16)) for nets [i] based on digital (0, 13, 257)-net over F256, using
- digital (26, 52, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 26, 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, 26, 257)-net over F256, using
- digital (12, 25, 257)-net over F16, using
(51, 77, 4132)-Net over F16 — Digital
Digital (51, 77, 4132)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1677, 4132, F16, 26) (dual of [4132, 4055, 27]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 20 times 0) [i] based on linear OA(1674, 4103, F16, 26) (dual of [4103, 4029, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [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(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(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(25) ⊂ Ce(23) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 20 times 0) [i] based on linear OA(1674, 4103, F16, 26) (dual of [4103, 4029, 27]-code), using
(51, 77, 5121646)-Net in Base 16 — Upper bound on s
There is no (51, 77, 5121647)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 521 481576 383099 834809 351075 671137 979219 904864 702322 957743 192544 802158 081119 604033 447177 671066 > 1677 [i]