Best Known (76, 76+39, s)-Nets in Base 16
(76, 76+39, 771)-Net over F16 — Constructive and digital
Digital (76, 115, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (18, 37, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(18,256) in PG(36,16)) for nets [i] based on digital (0, 19, 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(18,256) in PG(36,16)) for nets [i] based on digital (0, 19, 257)-net over F256, using
- digital (39, 78, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 39, 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, 39, 257)-net over F256, using
- digital (18, 37, 257)-net over F16, using
(76, 76+39, 4456)-Net over F16 — Digital
Digital (76, 115, 4456)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16115, 4456, F16, 39) (dual of [4456, 4341, 40]-code), using
- 347 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 28 times 0, 1, 94 times 0, 1, 216 times 0) [i] based on linear OA(16110, 4104, F16, 39) (dual of [4104, 3994, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(16109, 4097, F16, 39) (dual of [4097, 3988, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(16103, 4097, F16, 37) (dual of [4097, 3994, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [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 C([0,19]) ⊂ C([0,18]) [i] based on
- 347 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 28 times 0, 1, 94 times 0, 1, 216 times 0) [i] based on linear OA(16110, 4104, F16, 39) (dual of [4104, 3994, 40]-code), using
(76, 76+39, large)-Net in Base 16 — Upper bound on s
There is no (76, 115, large)-net in base 16, because
- 37 times m-reduction [i] would yield (76, 78, large)-net in base 16, but