Best Known (24, 31, s)-Nets in Base 16
(24, 31, 349526)-Net over F16 — Constructive and digital
Digital (24, 31, 349526)-net over F16, using
- net defined by OOA [i] based on linear OOA(1631, 349526, F16, 7, 7) (dual of [(349526, 7), 2446651, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1631, 1048579, F16, 7) (dual of [1048579, 1048548, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1631, 1048581, F16, 7) (dual of [1048581, 1048550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 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(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(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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(1631, 1048581, F16, 7) (dual of [1048581, 1048550, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1631, 1048579, F16, 7) (dual of [1048579, 1048548, 8]-code), using
(24, 31, 1048581)-Net over F16 — Digital
Digital (24, 31, 1048581)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1631, 1048581, F16, 7) (dual of [1048581, 1048550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 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(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(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(6) ⊂ Ce(5) [i] based on
(24, 31, large)-Net in Base 16 — Upper bound on s
There is no (24, 31, large)-net in base 16, because
- 5 times m-reduction [i] would yield (24, 26, large)-net in base 16, but