Best Known (89, 118, s)-Nets in Base 16
(89, 118, 9363)-Net over F16 — Constructive and digital
Digital (89, 118, 9363)-net over F16, using
- 162 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
(89, 118, 120259)-Net over F16 — Digital
Digital (89, 118, 120259)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16118, 120259, F16, 29) (dual of [120259, 120141, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16118, 131088, F16, 29) (dual of [131088, 130970, 30]-code), using
- trace code [i] based on linear OA(25659, 65544, F256, 29) (dual of [65544, 65485, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- 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(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(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- trace code [i] based on linear OA(25659, 65544, F256, 29) (dual of [65544, 65485, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16118, 131088, F16, 29) (dual of [131088, 130970, 30]-code), using
(89, 118, large)-Net in Base 16 — Upper bound on s
There is no (89, 118, large)-net in base 16, because
- 27 times m-reduction [i] would yield (89, 91, large)-net in base 16, but