Best Known (81−13, 81, s)-Nets in Base 4
(81−13, 81, 10934)-Net over F4 — Constructive and digital
Digital (68, 81, 10934)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (60, 73, 10922)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 10922, F4, 13, 13) (dual of [(10922, 13), 141913, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(473, 65533, F4, 13) (dual of [65533, 65460, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(473, 65533, F4, 13) (dual of [65533, 65460, 14]-code), using
- net defined by OOA [i] based on linear OOA(473, 10922, F4, 13, 13) (dual of [(10922, 13), 141913, 14]-NRT-code), using
(81−13, 81, 39122)-Net over F4 — Digital
Digital (68, 81, 39122)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(481, 39122, F4, 13) (dual of [39122, 39041, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 65537, F4, 13) (dual of [65537, 65456, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(481, 65537, F4, 13) (dual of [65537, 65456, 14]-code), using
(81−13, 81, large)-Net in Base 4 — Upper bound on s
There is no (68, 81, large)-net in base 4, because
- 11 times m-reduction [i] would yield (68, 70, large)-net in base 4, but