Best Known (12, 16, s)-Nets in Base 16
(12, 16, 524290)-Net over F16 — Constructive and digital
Digital (12, 16, 524290)-net over F16, using
- net defined by OOA [i] based on linear OOA(1616, 524290, F16, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1611, 1048576, F16, 3) (dual of [1048576, 1048565, 4]-code or 1048576-cap in PG(10,16)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
(12, 16, 1048581)-Net over F16 — Digital
Digital (12, 16, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1611, 1048576, F16, 3) (dual of [1048576, 1048565, 4]-code or 1048576-cap in PG(10,16)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
(12, 16, large)-Net in Base 16 — Upper bound on s
There is no (12, 16, large)-net in base 16, because
- 2 times m-reduction [i] would yield (12, 14, large)-net in base 16, but