Best Known (91, 91+33, s)-Nets in Base 16
(91, 91+33, 4096)-Net over F16 — Constructive and digital
Digital (91, 124, 4096)-net over F16, using
- 163 times duplication [i] based on digital (88, 121, 4096)-net over F16, using
- net defined by OOA [i] based on linear OOA(16121, 4096, F16, 33, 33) (dual of [(4096, 33), 135047, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(16121, 65537, F16, 33) (dual of [65537, 65416, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(16121, 65537, F16, 33) (dual of [65537, 65416, 34]-code), using
- net defined by OOA [i] based on linear OOA(16121, 4096, F16, 33, 33) (dual of [(4096, 33), 135047, 34]-NRT-code), using
(91, 91+33, 49595)-Net over F16 — Digital
Digital (91, 124, 49595)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16124, 49595, F16, 33) (dual of [49595, 49471, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16124, 65548, F16, 33) (dual of [65548, 65424, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(16121, 65537, F16, 33) (dual of [65537, 65416, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(16113, 65537, F16, 29) (dual of [65537, 65424, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 168−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(163, 11, F16, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,16) or 11-cap in PG(2,16)), using
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- Reed–Solomon code RS(13,16) [i]
- discarding factors / shortening the dual code based on linear OA(163, 16, F16, 3) (dual of [16, 13, 4]-code or 16-arc in PG(2,16) or 16-cap in PG(2,16)), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(16124, 65548, F16, 33) (dual of [65548, 65424, 34]-code), using
(91, 91+33, large)-Net in Base 16 — Upper bound on s
There is no (91, 124, large)-net in base 16, because
- 31 times m-reduction [i] would yield (91, 93, large)-net in base 16, but