Best Known (89, 89+30, s)-Nets in Base 16
(89, 89+30, 8738)-Net over F16 — Constructive and digital
Digital (89, 119, 8738)-net over F16, using
- 161 times duplication [i] based on digital (88, 118, 8738)-net over F16, using
- net defined by OOA [i] based on linear OOA(16118, 8738, F16, 30, 30) (dual of [(8738, 30), 262022, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(16118, 131070, F16, 30) (dual of [131070, 130952, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16118, 131072, F16, 30) (dual of [131072, 130954, 31]-code), using
- trace code [i] based on linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- trace code [i] based on linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16118, 131072, F16, 30) (dual of [131072, 130954, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(16118, 131070, F16, 30) (dual of [131070, 130952, 31]-code), using
- net defined by OOA [i] based on linear OOA(16118, 8738, F16, 30, 30) (dual of [(8738, 30), 262022, 31]-NRT-code), using
(89, 89+30, 89402)-Net over F16 — Digital
Digital (89, 119, 89402)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16119, 89402, F16, 30) (dual of [89402, 89283, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16119, 131077, F16, 30) (dual of [131077, 130958, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16118, 131076, F16, 30) (dual of [131076, 130958, 31]-code), using
- trace code [i] based on linear OA(25659, 65538, F256, 30) (dual of [65538, 65479, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(29) ⊂ Ce(28) [i] based on
- trace code [i] based on linear OA(25659, 65538, F256, 30) (dual of [65538, 65479, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16118, 131076, F16, 30) (dual of [131076, 130958, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16119, 131077, F16, 30) (dual of [131077, 130958, 31]-code), using
(89, 89+30, large)-Net in Base 16 — Upper bound on s
There is no (89, 119, large)-net in base 16, because
- 28 times m-reduction [i] would yield (89, 91, large)-net in base 16, but