Best Known (57−12, 57, s)-Nets in Base 16
(57−12, 57, 174764)-Net over F16 — Constructive and digital
Digital (45, 57, 174764)-net over F16, using
- net defined by OOA [i] based on linear OOA(1657, 174764, F16, 12, 12) (dual of [(174764, 12), 2097111, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1657, 1048584, F16, 12) (dual of [1048584, 1048527, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(1657, 1048587, F16, 12) (dual of [1048587, 1048530, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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(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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(1657, 1048587, F16, 12) (dual of [1048587, 1048530, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1657, 1048584, F16, 12) (dual of [1048584, 1048527, 13]-code), using
(57−12, 57, 1048587)-Net over F16 — Digital
Digital (45, 57, 1048587)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1657, 1048587, F16, 12) (dual of [1048587, 1048530, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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(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(11) ⊂ Ce(9) [i] based on
(57−12, 57, large)-Net in Base 16 — Upper bound on s
There is no (45, 57, large)-net in base 16, because
- 10 times m-reduction [i] would yield (45, 47, large)-net in base 16, but