Best Known (88, 88+33, s)-Nets in Base 16
(88, 88+33, 4096)-Net over F16 — Constructive and digital
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
(88, 88+33, 37919)-Net over F16 — Digital
Digital (88, 121, 37919)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16121, 37919, F16, 33) (dual of [37919, 37798, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16121, 65536, F16, 33) (dual of [65536, 65415, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(16121, 65536, F16, 33) (dual of [65536, 65415, 34]-code), using
(88, 88+33, large)-Net in Base 16 — Upper bound on s
There is no (88, 121, large)-net in base 16, because
- 31 times m-reduction [i] would yield (88, 90, large)-net in base 16, but