Best Known (104−23, 104, s)-Nets in Base 16
(104−23, 104, 11953)-Net over F16 — Constructive and digital
Digital (81, 104, 11953)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (67, 90, 11915)-net over F16, using
- net defined by OOA [i] based on linear OOA(1690, 11915, F16, 23, 23) (dual of [(11915, 23), 273955, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(1690, 131066, F16, 23) (dual of [131066, 130976, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 131074, F16, 23) (dual of [131074, 130984, 24]-code), using
- trace code [i] based on linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- trace code [i] based on linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 131074, F16, 23) (dual of [131074, 130984, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(1690, 131066, F16, 23) (dual of [131066, 130976, 24]-code), using
- net defined by OOA [i] based on linear OOA(1690, 11915, F16, 23, 23) (dual of [(11915, 23), 273955, 24]-NRT-code), using
- digital (3, 14, 38)-net over F16, using
(104−23, 104, 23832)-Net in Base 16 — Constructive
(81, 104, 23832)-net in base 16, using
- 161 times duplication [i] based on (80, 103, 23832)-net in base 16, using
- net defined by OOA [i] based on OOA(16103, 23832, S16, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16103, 262153, S16, 23), using
- 1 times code embedding in larger space [i] based on OA(16102, 262152, S16, 23), using
- discarding parts of the base [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- 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(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding parts of the base [i] based on linear OA(6468, 262152, F64, 23) (dual of [262152, 262084, 24]-code), using
- 1 times code embedding in larger space [i] based on OA(16102, 262152, S16, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16103, 262153, S16, 23), using
- net defined by OOA [i] based on OOA(16103, 23832, S16, 23, 23), using
(104−23, 104, 297154)-Net over F16 — Digital
Digital (81, 104, 297154)-net over F16, using
(104−23, 104, large)-Net in Base 16 — Upper bound on s
There is no (81, 104, large)-net in base 16, because
- 21 times m-reduction [i] would yield (81, 83, large)-net in base 16, but