Best Known (37, 37+11, s)-Nets in Base 16
(37, 37+11, 26239)-Net over F16 — Constructive and digital
Digital (37, 48, 26239)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (31, 42, 26215)-net over F16, using
- net defined by OOA [i] based on linear OOA(1642, 26215, F16, 11, 11) (dual of [(26215, 11), 288323, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1642, 131076, F16, 11) (dual of [131076, 131034, 12]-code), using
- trace code [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1642, 131076, F16, 11) (dual of [131076, 131034, 12]-code), using
- net defined by OOA [i] based on linear OOA(1642, 26215, F16, 11, 11) (dual of [(26215, 11), 288323, 12]-NRT-code), using
- digital (1, 6, 24)-net over F16, using
(37, 37+11, 52430)-Net in Base 16 — Constructive
(37, 48, 52430)-net in base 16, using
- base change [i] based on digital (21, 32, 52430)-net over F64, using
- net defined by OOA [i] based on linear OOA(6432, 52430, F64, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(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(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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
- net defined by OOA [i] based on linear OOA(6432, 52430, F64, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
(37, 37+11, 181833)-Net over F16 — Digital
Digital (37, 48, 181833)-net over F16, using
(37, 37+11, large)-Net in Base 16 — Upper bound on s
There is no (37, 48, large)-net in base 16, because
- 9 times m-reduction [i] would yield (37, 39, large)-net in base 16, but