Best Known (46, 59, s)-Nets in Base 16
(46, 59, 21883)-Net over F16 — Constructive and digital
Digital (46, 59, 21883)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 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 (37, 50, 21845)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- digital (3, 9, 38)-net over F16, using
(46, 59, 43692)-Net in Base 16 — Constructive
(46, 59, 43692)-net in base 16, using
- 161 times duplication [i] based on (45, 58, 43692)-net in base 16, using
- net defined by OOA [i] based on OOA(1658, 43692, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1658, 262153, S16, 13), using
- 1 times code embedding in larger space [i] based on OA(1657, 262152, S16, 13), using
- discarding parts of the base [i] based on linear OA(6438, 262152, F64, 13) (dual of [262152, 262114, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(6438, 262152, F64, 13) (dual of [262152, 262114, 14]-code), using
- 1 times code embedding in larger space [i] based on OA(1657, 262152, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1658, 262153, S16, 13), using
- net defined by OOA [i] based on OOA(1658, 43692, S16, 13, 13), using
(46, 59, 293451)-Net over F16 — Digital
Digital (46, 59, 293451)-net over F16, using
(46, 59, large)-Net in Base 16 — Upper bound on s
There is no (46, 59, large)-net in base 16, because
- 11 times m-reduction [i] would yield (46, 48, large)-net in base 16, but