Best Known (39−9, 39, s)-Nets in Base 16
(39−9, 39, 32792)-Net over F16 — Constructive and digital
Digital (30, 39, 32792)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (25, 34, 32768)-net over F16, using
- net defined by OOA [i] based on linear OOA(1634, 32768, F16, 9, 9) (dual of [(32768, 9), 294878, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1634, 131073, F16, 9) (dual of [131073, 131039, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- trace code [i] based on linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(1634, 131074, F16, 9) (dual of [131074, 131040, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1634, 131073, F16, 9) (dual of [131073, 131039, 10]-code), using
- net defined by OOA [i] based on linear OOA(1634, 32768, F16, 9, 9) (dual of [(32768, 9), 294878, 10]-NRT-code), using
- digital (1, 5, 24)-net over F16, using
(39−9, 39, 65537)-Net in Base 16 — Constructive
(30, 39, 65537)-net in base 16, using
- base change [i] based on digital (17, 26, 65537)-net over F64, using
- net defined by OOA [i] based on linear OOA(6426, 65537, F64, 9, 9) (dual of [(65537, 9), 589807, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6426, 262149, F64, 9) (dual of [262149, 262123, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(6419, 262145, F64, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 262152, F64, 9) (dual of [262152, 262126, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6426, 262149, F64, 9) (dual of [262149, 262123, 10]-code), using
- net defined by OOA [i] based on linear OOA(6426, 65537, F64, 9, 9) (dual of [(65537, 9), 589807, 10]-NRT-code), using
(39−9, 39, 186077)-Net over F16 — Digital
Digital (30, 39, 186077)-net over F16, using
(39−9, 39, large)-Net in Base 16 — Upper bound on s
There is no (30, 39, large)-net in base 16, because
- 7 times m-reduction [i] would yield (30, 32, large)-net in base 16, but