Best Known (82, 82+44, s)-Nets in Base 16
(82, 82+44, 589)-Net over F16 — Constructive and digital
Digital (82, 126, 589)-net over F16, using
- (u, u+v)-construction [i] based on
- 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 (54, 98, 524)-net over F16, using
- trace code for nets [i] based on digital (5, 49, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 49, 262)-net over F256, using
- digital (6, 28, 65)-net over F16, using
(82, 82+44, 618)-Net in Base 16 — Constructive
(82, 126, 618)-net in base 16, using
- (u, u+v)-construction [i] based on
- (16, 38, 104)-net in base 16, using
- 1 times m-reduction [i] based on (16, 39, 104)-net in base 16, using
- base change [i] based on digital (3, 26, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- base change [i] based on digital (3, 26, 104)-net over F64, using
- 1 times m-reduction [i] based on (16, 39, 104)-net in base 16, using
- digital (44, 88, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 44, 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, 44, 257)-net over F256, using
- (16, 38, 104)-net in base 16, using
(82, 82+44, 4113)-Net over F16 — Digital
Digital (82, 126, 4113)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16126, 4113, F16, 44) (dual of [4113, 3987, 45]-code), using
- 12 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0) [i] based on linear OA(16124, 4099, F16, 44) (dual of [4099, 3975, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(42) [i] based on
- 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(16121, 4096, F16, 43) (dual of [4096, 3975, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(43) ⊂ Ce(42) [i] based on
- 12 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0) [i] based on linear OA(16124, 4099, F16, 44) (dual of [4099, 3975, 45]-code), using
(82, 82+44, 4754274)-Net in Base 16 — Upper bound on s
There is no (82, 126, 4754275)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 52 374480 823006 892813 981755 597344 846137 136620 909712 309016 584166 351392 923242 855963 056520 206604 618877 244532 872946 318542 744944 525717 971484 180762 410311 546376 > 16126 [i]