Best Known (49, 49+12, s)-Nets in Base 16
(49, 49+12, 174767)-Net over F16 — Constructive and digital
Digital (49, 61, 174767)-net over F16, using
- net defined by OOA [i] based on linear OOA(1661, 174767, F16, 12, 12) (dual of [(174767, 12), 2097143, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1661, 1048602, F16, 12) (dual of [1048602, 1048541, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(7) ⊂ Ce(6) [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(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(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(164, 25, F16, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,16)), using
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(11) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- OA 6-folding and stacking [i] based on linear OA(1661, 1048602, F16, 12) (dual of [1048602, 1048541, 13]-code), using
(49, 49+12, 349526)-Net in Base 16 — Constructive
(49, 61, 349526)-net in base 16, using
- net defined by OOA [i] based on OOA(1661, 349526, S16, 12, 12), using
- OA 6-folding and stacking [i] based on OA(1661, 2097156, S16, 12), using
- 1 times code embedding in larger space [i] based on OA(1660, 2097155, S16, 12), using
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- 1 times code embedding in larger space [i] based on OA(1660, 2097155, S16, 12), using
- OA 6-folding and stacking [i] based on OA(1661, 2097156, S16, 12), using
(49, 49+12, 1557102)-Net over F16 — Digital
Digital (49, 61, 1557102)-net over F16, using
(49, 49+12, large)-Net in Base 16 — Upper bound on s
There is no (49, 61, large)-net in base 16, because
- 10 times m-reduction [i] would yield (49, 51, large)-net in base 16, but