Best Known (107−23, 107, s)-Nets in Base 16
(107−23, 107, 95326)-Net over F16 — Constructive and digital
Digital (84, 107, 95326)-net over F16, using
- net defined by OOA [i] based on linear OOA(16107, 95326, F16, 23, 23) (dual of [(95326, 23), 2192391, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16107, 1048587, F16, 23) (dual of [1048587, 1048480, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(16107, 1048587, F16, 23) (dual of [1048587, 1048480, 24]-code), using
(107−23, 107, 692362)-Net over F16 — Digital
Digital (84, 107, 692362)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16107, 692362, F16, 23) (dual of [692362, 692255, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(16107, 1048587, F16, 23) (dual of [1048587, 1048480, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(16107, 1048587, F16, 23) (dual of [1048587, 1048480, 24]-code), using
(107−23, 107, large)-Net in Base 16 — Upper bound on s
There is no (84, 107, large)-net in base 16, because
- 21 times m-reduction [i] would yield (84, 86, large)-net in base 16, but