Best Known (89, 89+28, s)-Nets in Base 16
(89, 89+28, 9363)-Net over F16 — Constructive and digital
Digital (89, 117, 9363)-net over F16, using
- 161 times duplication [i] based on digital (88, 116, 9363)-net over F16, using
- t-expansion [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
- t-expansion [i] based on digital (87, 116, 9363)-net over F16, using
(89, 89+28, 131096)-Net over F16 — Digital
Digital (89, 117, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16117, 131096, F16, 28) (dual of [131096, 130979, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16116, 131094, F16, 28) (dual of [131094, 130978, 29]-code), using
- trace code [i] based on linear OA(25658, 65547, F256, 28) (dual of [65547, 65489, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [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(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(25658, 65547, F256, 28) (dual of [65547, 65489, 29]-code), using
- linear OA(16116, 131095, F16, 27) (dual of [131095, 130979, 28]-code), using Gilbert–Varšamov bound and bm = 16116 > Vbs−1(k−1) = 106865 100410 059336 493689 566262 774517 037532 435818 553780 201088 950113 998220 340500 699264 034343 206138 276068 484675 062726 708095 154619 255704 027136 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16116, 131094, F16, 28) (dual of [131094, 130978, 29]-code), using
- construction X with Varšamov bound [i] based on
(89, 89+28, large)-Net in Base 16 — Upper bound on s
There is no (89, 117, large)-net in base 16, because
- 26 times m-reduction [i] would yield (89, 91, large)-net in base 16, but