Best Known (122−32, 122, s)-Nets in Base 16
(122−32, 122, 4096)-Net over F16 — Constructive and digital
Digital (90, 122, 4096)-net over F16, using
- 161 times duplication [i] based on digital (89, 121, 4096)-net over F16, using
- t-expansion [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
- t-expansion [i] based on digital (88, 121, 4096)-net over F16, using
(122−32, 122, 57702)-Net over F16 — Digital
Digital (90, 122, 57702)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16122, 57702, F16, 32) (dual of [57702, 57580, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(16122, 65545, F16, 32) (dual of [65545, 65423, 33]-code), using
- 1 times truncation [i] based on linear OA(16123, 65546, F16, 33) (dual of [65546, 65423, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] 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]
- linear OA(16113, 65536, F16, 30) (dual of [65536, 65423, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(162, 10, F16, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- 1 times truncation [i] based on linear OA(16123, 65546, F16, 33) (dual of [65546, 65423, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(16122, 65545, F16, 32) (dual of [65545, 65423, 33]-code), using
(122−32, 122, large)-Net in Base 16 — Upper bound on s
There is no (90, 122, large)-net in base 16, because
- 30 times m-reduction [i] would yield (90, 92, large)-net in base 16, but