Best Known (68, 68+19, s)-Nets in Base 16
(68, 68+19, 116509)-Net over F16 — Constructive and digital
Digital (68, 87, 116509)-net over F16, using
- net defined by OOA [i] based on linear OOA(1687, 116509, F16, 19, 19) (dual of [(116509, 19), 2213584, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1687, 1048582, F16, 19) (dual of [1048582, 1048495, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1687, 1048587, F16, 19) (dual of [1048587, 1048500, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1687, 1048587, F16, 19) (dual of [1048587, 1048500, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1687, 1048582, F16, 19) (dual of [1048582, 1048495, 20]-code), using
(68, 68+19, 590579)-Net over F16 — Digital
Digital (68, 87, 590579)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1687, 590579, F16, 19) (dual of [590579, 590492, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1687, 1048587, F16, 19) (dual of [1048587, 1048500, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1687, 1048587, F16, 19) (dual of [1048587, 1048500, 20]-code), using
(68, 68+19, large)-Net in Base 16 — Upper bound on s
There is no (68, 87, large)-net in base 16, because
- 17 times m-reduction [i] would yield (68, 70, large)-net in base 16, but