Best Known (110−38, 110, s)-Nets in Base 16
(110−38, 110, 596)-Net over F16 — Constructive and digital
Digital (72, 110, 596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 34, 82)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (6, 25, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (0, 9, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- 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, 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, 38, 257)-net over F256, using
- digital (15, 34, 82)-net over F16, using
(110−38, 110, 643)-Net in Base 16 — Constructive
(72, 110, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (15, 34, 129)-net in base 16, using
- 1 times m-reduction [i] based on (15, 35, 129)-net in base 16, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
- 1 times m-reduction [i] based on (15, 35, 129)-net in base 16, 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, 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, 38, 257)-net over F256, using
- (15, 34, 129)-net in base 16, using
(110−38, 110, 4129)-Net over F16 — Digital
Digital (72, 110, 4129)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16110, 4129, F16, 38) (dual of [4129, 4019, 39]-code), using
- 23 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 18 times 0) [i] based on linear OA(16107, 4103, F16, 38) (dual of [4103, 3996, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(35) [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(16100, 4096, F16, 36) (dual of [4096, 3996, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(37) ⊂ Ce(35) [i] based on
- 23 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 18 times 0) [i] based on linear OA(16107, 4103, F16, 38) (dual of [4103, 3996, 39]-code), using
(110−38, 110, 4947027)-Net in Base 16 — Upper bound on s
There is no (72, 110, 4947028)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 2 839222 488836 406473 944215 985375 594819 909719 996350 669380 040996 428146 920971 038169 646818 079421 099493 971314 373905 372800 513636 032024 921331 > 16110 [i]