Best Known (37, 37+10, s)-Nets in Base 16
(37, 37+10, 209717)-Net over F16 — Constructive and digital
Digital (37, 47, 209717)-net over F16, using
- net defined by OOA [i] based on linear OOA(1647, 209717, F16, 10, 10) (dual of [(209717, 10), 2097123, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1647, 1048585, F16, 10) (dual of [1048585, 1048538, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1647, 1048587, F16, 10) (dual of [1048587, 1048540, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(1647, 1048587, F16, 10) (dual of [1048587, 1048540, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1647, 1048585, F16, 10) (dual of [1048585, 1048538, 11]-code), using
(37, 37+10, 1048588)-Net over F16 — Digital
Digital (37, 47, 1048588)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1647, 1048588, F16, 10) (dual of [1048588, 1048541, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1611, 12, F16, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,16)), using
- dual of repetition code with length 12 [i]
- linear OA(161, 12, F16, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
(37, 37+10, large)-Net in Base 16 — Upper bound on s
There is no (37, 47, large)-net in base 16, because
- 8 times m-reduction [i] would yield (37, 39, large)-net in base 16, but