Best Known (83, 83+27, s)-Nets in Base 16
(83, 83+27, 10083)-Net over F16 — Constructive and digital
Digital (83, 110, 10083)-net over F16, using
- 162 times duplication [i] based on digital (81, 108, 10083)-net over F16, using
- net defined by OOA [i] based on linear OOA(16108, 10083, F16, 27, 27) (dual of [(10083, 27), 272133, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16108, 131080, F16, 27) (dual of [131080, 130972, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16108, 131084, F16, 27) (dual of [131084, 130976, 28]-code), using
- trace code [i] based on linear OA(25654, 65542, F256, 27) (dual of [65542, 65488, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- 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(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- trace code [i] based on linear OA(25654, 65542, F256, 27) (dual of [65542, 65488, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16108, 131084, F16, 27) (dual of [131084, 130976, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16108, 131080, F16, 27) (dual of [131080, 130972, 28]-code), using
- net defined by OOA [i] based on linear OOA(16108, 10083, F16, 27, 27) (dual of [(10083, 27), 272133, 28]-NRT-code), using
(83, 83+27, 120629)-Net over F16 — Digital
Digital (83, 110, 120629)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16110, 120629, F16, 27) (dual of [120629, 120519, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 131088, F16, 27) (dual of [131088, 130978, 28]-code), using
- trace code [i] based on linear OA(25655, 65544, F256, 27) (dual of [65544, 65489, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- 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(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(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(26) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(25655, 65544, F256, 27) (dual of [65544, 65489, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 131088, F16, 27) (dual of [131088, 130978, 28]-code), using
(83, 83+27, large)-Net in Base 16 — Upper bound on s
There is no (83, 110, large)-net in base 16, because
- 25 times m-reduction [i] would yield (83, 85, large)-net in base 16, but