Best Known (117−29, 117, s)-Nets in Base 16
(117−29, 117, 9363)-Net over F16 — Constructive and digital
Digital (88, 117, 9363)-net over F16, using
- 161 times duplication [i] based on digital (87, 116, 9363)-net over F16, using
- net defined by OOA [i] based on linear OOA(16116, 9363, F16, 29, 29) (dual of [(9363, 29), 271411, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16116, 131083, F16, 29) (dual of [131083, 130967, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16116, 131084, F16, 29) (dual of [131084, 130968, 30]-code), using
- trace code [i] based on linear OA(25658, 65542, F256, 29) (dual of [65542, 65484, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [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 C([0,14]) ⊂ C([0,13]) [i] based on
- trace code [i] based on linear OA(25658, 65542, F256, 29) (dual of [65542, 65484, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16116, 131084, F16, 29) (dual of [131084, 130968, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16116, 131083, F16, 29) (dual of [131083, 130967, 30]-code), using
- net defined by OOA [i] based on linear OOA(16116, 9363, F16, 29, 29) (dual of [(9363, 29), 271411, 30]-NRT-code), using
(117−29, 117, 108521)-Net over F16 — Digital
Digital (88, 117, 108521)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16117, 108521, F16, 29) (dual of [108521, 108404, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16117, 131085, F16, 29) (dual of [131085, 130968, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16116, 131084, F16, 29) (dual of [131084, 130968, 30]-code), using
- trace code [i] based on linear OA(25658, 65542, F256, 29) (dual of [65542, 65484, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [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 C([0,14]) ⊂ C([0,13]) [i] based on
- trace code [i] based on linear OA(25658, 65542, F256, 29) (dual of [65542, 65484, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16116, 131084, F16, 29) (dual of [131084, 130968, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16117, 131085, F16, 29) (dual of [131085, 130968, 30]-code), using
(117−29, 117, large)-Net in Base 16 — Upper bound on s
There is no (88, 117, large)-net in base 16, because
- 27 times m-reduction [i] would yield (88, 90, large)-net in base 16, but