Best Known (68−15, 68, s)-Nets in Base 16
(68−15, 68, 18763)-Net over F16 — Constructive and digital
Digital (53, 68, 18763)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 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 (43, 58, 18725)-net over F16, using
- net defined by OOA [i] based on linear OOA(1658, 18725, F16, 15, 15) (dual of [(18725, 15), 280817, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1658, 131076, F16, 15) (dual of [131076, 131018, 16]-code), using
- trace code [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1658, 131076, F16, 15) (dual of [131076, 131018, 16]-code), using
- net defined by OOA [i] based on linear OOA(1658, 18725, F16, 15, 15) (dual of [(18725, 15), 280817, 16]-NRT-code), using
- digital (3, 10, 38)-net over F16, using
(68−15, 68, 37450)-Net in Base 16 — Constructive
(53, 68, 37450)-net in base 16, using
- 162 times duplication [i] based on (51, 66, 37450)-net in base 16, using
- base change [i] based on digital (29, 44, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- base change [i] based on digital (29, 44, 37450)-net over F64, using
(68−15, 68, 284420)-Net over F16 — Digital
Digital (53, 68, 284420)-net over F16, using
(68−15, 68, large)-Net in Base 16 — Upper bound on s
There is no (53, 68, large)-net in base 16, because
- 13 times m-reduction [i] would yield (53, 55, large)-net in base 16, but