Best Known (61, 81, s)-Nets in Base 16
(61, 81, 13108)-Net over F16 — Constructive and digital
Digital (61, 81, 13108)-net over F16, using
- 161 times duplication [i] based on digital (60, 80, 13108)-net over F16, using
- net defined by OOA [i] based on linear OOA(1680, 13108, F16, 20, 20) (dual of [(13108, 20), 262080, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1680, 131080, F16, 20) (dual of [131080, 131000, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 131082, F16, 20) (dual of [131082, 131002, 21]-code), using
- trace code [i] based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 131082, F16, 20) (dual of [131082, 131002, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1680, 131080, F16, 20) (dual of [131080, 131000, 21]-code), using
- net defined by OOA [i] based on linear OOA(1680, 13108, F16, 20, 20) (dual of [(13108, 20), 262080, 21]-NRT-code), using
(61, 81, 113148)-Net over F16 — Digital
Digital (61, 81, 113148)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1681, 113148, F16, 20) (dual of [113148, 113067, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1681, 131083, F16, 20) (dual of [131083, 131002, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1680, 131082, F16, 20) (dual of [131082, 131002, 21]-code), using
- trace code [i] based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(25640, 65541, F256, 20) (dual of [65541, 65501, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1680, 131082, F16, 20) (dual of [131082, 131002, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1681, 131083, F16, 20) (dual of [131083, 131002, 21]-code), using
(61, 81, large)-Net in Base 16 — Upper bound on s
There is no (61, 81, large)-net in base 16, because
- 18 times m-reduction [i] would yield (61, 63, large)-net in base 16, but