Best Known (84, 129, s)-Nets in Base 16
(84, 129, 596)-Net over F16 — Constructive and digital
Digital (84, 129, 596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (17, 39, 82)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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, 28, 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, 11, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (45, 90, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 45, 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, 45, 257)-net over F256, using
- digital (17, 39, 82)-net over F16, using
(84, 129, 643)-Net in Base 16 — Constructive
(84, 129, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (17, 39, 129)-net in base 16, using
- base change [i] based on (4, 26, 129)-net in base 64, using
- 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 129)-net over F128, using
- 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on (4, 26, 129)-net in base 64, using
- digital (45, 90, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 45, 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, 45, 257)-net over F256, using
- (17, 39, 129)-net in base 16, using
(84, 129, 4115)-Net over F16 — Digital
Digital (84, 129, 4115)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16129, 4115, F16, 45) (dual of [4115, 3986, 46]-code), using
- 14 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0) [i] based on linear OA(16127, 4099, F16, 45) (dual of [4099, 3972, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(43) [i] based on
- linear OA(16127, 4096, F16, 45) (dual of [4096, 3969, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(16124, 4096, F16, 44) (dual of [4096, 3972, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [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(44) ⊂ Ce(43) [i] based on
- 14 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0) [i] based on linear OA(16127, 4099, F16, 45) (dual of [4099, 3972, 46]-code), using
(84, 129, 6117160)-Net in Base 16 — Upper bound on s
There is no (84, 129, 6117161)-net in base 16, because
- 1 times m-reduction [i] would yield (84, 128, 6117161)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 13407 809785 435751 774624 890224 279290 236230 701792 805513 458372 924061 691978 025674 192791 021669 704818 147371 138361 177978 329832 917563 423556 479084 609834 448962 922856 > 16128 [i]