Best Known (77, 77+26, s)-Nets in Base 16
(77, 77+26, 10082)-Net over F16 — Constructive and digital
Digital (77, 103, 10082)-net over F16, using
- 161 times duplication [i] based on digital (76, 102, 10082)-net over F16, using
- net defined by OOA [i] based on linear OOA(16102, 10082, F16, 26, 26) (dual of [(10082, 26), 262030, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(16102, 131066, F16, 26) (dual of [131066, 130964, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16102, 131072, F16, 26) (dual of [131072, 130970, 27]-code), using
- trace code [i] based on linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- trace code [i] based on linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16102, 131072, F16, 26) (dual of [131072, 130970, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(16102, 131066, F16, 26) (dual of [131066, 130964, 27]-code), using
- net defined by OOA [i] based on linear OOA(16102, 10082, F16, 26, 26) (dual of [(10082, 26), 262030, 27]-NRT-code), using
(77, 77+26, 85648)-Net over F16 — Digital
Digital (77, 103, 85648)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16103, 85648, F16, 26) (dual of [85648, 85545, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16103, 131077, F16, 26) (dual of [131077, 130974, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16102, 131076, F16, 26) (dual of [131076, 130974, 27]-code), using
- trace code [i] based on linear OA(25651, 65538, F256, 26) (dual of [65538, 65487, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- trace code [i] based on linear OA(25651, 65538, F256, 26) (dual of [65538, 65487, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16102, 131076, F16, 26) (dual of [131076, 130974, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16103, 131077, F16, 26) (dual of [131077, 130974, 27]-code), using
(77, 77+26, large)-Net in Base 16 — Upper bound on s
There is no (77, 103, large)-net in base 16, because
- 24 times m-reduction [i] would yield (77, 79, large)-net in base 16, but