Best Known (82−11, 82, s)-Nets in Base 5
(82−11, 82, 1677720)-Net over F5 — Constructive and digital
Digital (71, 82, 1677720)-net over F5, using
- 51 times duplication [i] based on digital (70, 81, 1677720)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
(82−11, 82, 4194301)-Net over F5 — Digital
Digital (71, 82, 4194301)-net over F5, using
- 51 times duplication [i] based on digital (70, 81, 4194301)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(581, 4194301, F5, 2, 11) (dual of [(4194301, 2), 8388521, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(581, 8388602, F5, 11) (dual of [8388602, 8388521, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 2-folding [i] based on linear OA(581, 8388602, F5, 11) (dual of [8388602, 8388521, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(581, 4194301, F5, 2, 11) (dual of [(4194301, 2), 8388521, 12]-NRT-code), using
(82−11, 82, large)-Net in Base 5 — Upper bound on s
There is no (71, 82, large)-net in base 5, because
- 9 times m-reduction [i] would yield (71, 73, large)-net in base 5, but