Best Known (75, 113, s)-Nets in Base 16
(75, 113, 771)-Net over F16 — Constructive and digital
Digital (75, 113, 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 (38, 76, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 38, 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, 38, 257)-net over F256, using
- digital (18, 37, 257)-net over F16, using
(75, 113, 4673)-Net over F16 — Digital
Digital (75, 113, 4673)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16113, 4673, F16, 38) (dual of [4673, 4560, 39]-code), using
- 567 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 17 times 0, 1, 67 times 0, 1, 178 times 0, 1, 296 times 0) [i] based on linear OA(16106, 4099, F16, 38) (dual of [4099, 3993, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(16106, 4096, F16, 38) (dual of [4096, 3990, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(16103, 4096, F16, 37) (dual of [4096, 3993, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 567 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 17 times 0, 1, 67 times 0, 1, 178 times 0, 1, 296 times 0) [i] based on linear OA(16106, 4099, F16, 38) (dual of [4099, 3993, 39]-code), using
(75, 113, 7664235)-Net in Base 16 — Upper bound on s
There is no (75, 113, 7664236)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 11629 441803 892005 107726 406542 777306 195914 349558 987080 570191 069975 403082 378257 888738 047382 307821 789023 011410 637535 250817 681890 047551 351986 > 16113 [i]