Best Known (64, 64+18, s)-Nets in Base 16
(64, 64+18, 116509)-Net over F16 — Constructive and digital
Digital (64, 82, 116509)-net over F16, using
- 161 times duplication [i] based on digital (63, 81, 116509)-net over F16, using
- net defined by OOA [i] based on linear OOA(1681, 116509, F16, 18, 18) (dual of [(116509, 18), 2097081, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(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(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- net defined by OOA [i] based on linear OOA(1681, 116509, F16, 18, 18) (dual of [(116509, 18), 2097081, 19]-NRT-code), using
(64, 64+18, 565326)-Net over F16 — Digital
Digital (64, 82, 565326)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1682, 565326, F16, 18) (dual of [565326, 565244, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 1048582, F16, 18) (dual of [1048582, 1048500, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(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(17) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 1048582, F16, 18) (dual of [1048582, 1048500, 19]-code), using
(64, 64+18, large)-Net in Base 16 — Upper bound on s
There is no (64, 82, large)-net in base 16, because
- 16 times m-reduction [i] would yield (64, 66, large)-net in base 16, but