Best Known (57, 74, s)-Nets in Base 16
(57, 74, 16401)-Net over F16 — Constructive and digital
Digital (57, 74, 16401)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (49, 66, 16384)-net over F16, using
- net defined by OOA [i] based on linear OOA(1666, 16384, F16, 17, 17) (dual of [(16384, 17), 278462, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1666, 131073, F16, 17) (dual of [131073, 131007, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1666, 131074, F16, 17) (dual of [131074, 131008, 18]-code), using
- trace code [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- trace code [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1666, 131074, F16, 17) (dual of [131074, 131008, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1666, 131073, F16, 17) (dual of [131073, 131007, 18]-code), using
- net defined by OOA [i] based on linear OOA(1666, 16384, F16, 17, 17) (dual of [(16384, 17), 278462, 18]-NRT-code), using
- digital (0, 8, 17)-net over F16, using
(57, 74, 32768)-Net in Base 16 — Constructive
(57, 74, 32768)-net in base 16, using
- net defined by OOA [i] based on OOA(1674, 32768, S16, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1674, 262145, S16, 17), using
- discarding factors based on OA(1674, 262147, S16, 17), using
- discarding parts of the base [i] based on linear OA(6449, 262147, F64, 17) (dual of [262147, 262098, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(6449, 262144, F64, 17) (dual of [262144, 262095, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(6449, 262147, F64, 17) (dual of [262147, 262098, 18]-code), using
- discarding factors based on OA(1674, 262147, S16, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1674, 262145, S16, 17), using
(57, 74, 168083)-Net over F16 — Digital
Digital (57, 74, 168083)-net over F16, using
(57, 74, large)-Net in Base 16 — Upper bound on s
There is no (57, 74, large)-net in base 16, because
- 15 times m-reduction [i] would yield (57, 59, large)-net in base 16, but