Best Known (83, 83+28, s)-Nets in Base 16
(83, 83+28, 9362)-Net over F16 — Constructive and digital
Digital (83, 111, 9362)-net over F16, using
- 161 times duplication [i] based on digital (82, 110, 9362)-net over F16, using
- net defined by OOA [i] based on linear OOA(16110, 9362, F16, 28, 28) (dual of [(9362, 28), 262026, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(16110, 131068, F16, 28) (dual of [131068, 130958, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 131072, F16, 28) (dual of [131072, 130962, 29]-code), using
- trace code [i] based on linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- trace code [i] based on linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 131072, F16, 28) (dual of [131072, 130962, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(16110, 131068, F16, 28) (dual of [131068, 130958, 29]-code), using
- net defined by OOA [i] based on linear OOA(16110, 9362, F16, 28, 28) (dual of [(9362, 28), 262026, 29]-NRT-code), using
(83, 83+28, 87395)-Net over F16 — Digital
Digital (83, 111, 87395)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16111, 87395, F16, 28) (dual of [87395, 87284, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16111, 131077, F16, 28) (dual of [131077, 130966, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16110, 131076, F16, 28) (dual of [131076, 130966, 29]-code), using
- trace code [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- trace code [i] based on linear OA(25655, 65538, F256, 28) (dual of [65538, 65483, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16110, 131076, F16, 28) (dual of [131076, 130966, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16111, 131077, F16, 28) (dual of [131077, 130966, 29]-code), using
(83, 83+28, large)-Net in Base 16 — Upper bound on s
There is no (83, 111, large)-net in base 16, because
- 26 times m-reduction [i] would yield (83, 85, large)-net in base 16, but