Best Known (28−6, 28, s)-Nets in Base 16
(28−6, 28, 349531)-Net over F16 — Constructive and digital
Digital (22, 28, 349531)-net over F16, using
- net defined by OOA [i] based on linear OOA(1628, 349531, F16, 6, 6) (dual of [(349531, 6), 2097158, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1628, 1048593, F16, 6) (dual of [1048593, 1048565, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(1626, 1048576, F16, 6) (dual of [1048576, 1048550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(1628, 1048593, F16, 6) (dual of [1048593, 1048565, 7]-code), using
(28−6, 28, 699051)-Net in Base 16 — Constructive
(22, 28, 699051)-net in base 16, using
- base change [i] based on digital (10, 16, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), using
(28−6, 28, 1048593)-Net over F16 — Digital
Digital (22, 28, 1048593)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1628, 1048593, F16, 6) (dual of [1048593, 1048565, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(1626, 1048576, F16, 6) (dual of [1048576, 1048550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(28−6, 28, 1390869)-Net in Base 16
(22, 28, 1390869)-net in base 16, using
- base change [i] based on digital (10, 16, 1390869)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
(28−6, 28, large)-Net in Base 16 — Upper bound on s
There is no (22, 28, large)-net in base 16, because
- 4 times m-reduction [i] would yield (22, 24, large)-net in base 16, but