Best Known (125−27, 125, s)-Nets in Base 16
(125−27, 125, 10147)-Net over F16 — Constructive and digital
Digital (98, 125, 10147)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 19, 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 (79, 106, 10082)-net over F16, using
- net defined by OOA [i] based on linear OOA(16106, 10082, F16, 27, 27) (dual of [(10082, 27), 272108, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16106, 131067, F16, 27) (dual of [131067, 130961, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16106, 131074, F16, 27) (dual of [131074, 130968, 28]-code), using
- trace code [i] based on linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- trace code [i] based on linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16106, 131074, F16, 27) (dual of [131074, 130968, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16106, 131067, F16, 27) (dual of [131067, 130961, 28]-code), using
- net defined by OOA [i] based on linear OOA(16106, 10082, F16, 27, 27) (dual of [(10082, 27), 272108, 28]-NRT-code), using
- digital (6, 19, 65)-net over F16, using
(125−27, 125, 20166)-Net in Base 16 — Constructive
(98, 125, 20166)-net in base 16, using
- 162 times duplication [i] based on (96, 123, 20166)-net in base 16, using
- base change [i] based on digital (55, 82, 20166)-net over F64, using
- net defined by OOA [i] based on linear OOA(6482, 20166, F64, 27, 27) (dual of [(20166, 27), 544400, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6482, 262159, F64, 27) (dual of [262159, 262077, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6482, 262160, F64, 27) (dual of [262160, 262078, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(6479, 262145, F64, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6482, 262160, F64, 27) (dual of [262160, 262078, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6482, 262159, F64, 27) (dual of [262159, 262077, 28]-code), using
- net defined by OOA [i] based on linear OOA(6482, 20166, F64, 27, 27) (dual of [(20166, 27), 544400, 28]-NRT-code), using
- base change [i] based on digital (55, 82, 20166)-net over F64, using
(125−27, 125, 432766)-Net over F16 — Digital
Digital (98, 125, 432766)-net over F16, using
(125−27, 125, large)-Net in Base 16 — Upper bound on s
There is no (98, 125, large)-net in base 16, because
- 25 times m-reduction [i] would yield (98, 100, large)-net in base 16, but