Best Known (54, 54+15, s)-Nets in Base 16
(54, 54+15, 18770)-Net over F16 — Constructive and digital
Digital (54, 69, 18770)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-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 (4, 11, 45)-net over F16, using
(54, 54+15, 37451)-Net in Base 16 — Constructive
(54, 69, 37451)-net in base 16, using
- base change [i] based on digital (31, 46, 37451)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 37451, F64, 15, 15) (dual of [(37451, 15), 561719, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6446, 262158, F64, 15) (dual of [262158, 262112, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [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(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(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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6446, 262158, F64, 15) (dual of [262158, 262112, 16]-code), using
- net defined by OOA [i] based on linear OOA(6446, 37451, F64, 15, 15) (dual of [(37451, 15), 561719, 16]-NRT-code), using
(54, 54+15, 346710)-Net over F16 — Digital
Digital (54, 69, 346710)-net over F16, using
(54, 54+15, large)-Net in Base 16 — Upper bound on s
There is no (54, 69, large)-net in base 16, because
- 13 times m-reduction [i] would yield (54, 56, large)-net in base 16, but