Best Known (162−24, 162, s)-Nets in Base 4
(162−24, 162, 21845)-Net over F4 — Constructive and digital
Digital (138, 162, 21845)-net over F4, using
- net defined by OOA [i] based on linear OOA(4162, 21845, F4, 24, 24) (dual of [(21845, 24), 524118, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4162, 262140, F4, 24) (dual of [262140, 261978, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4162, 262140, F4, 24) (dual of [262140, 261978, 25]-code), using
(162−24, 162, 111797)-Net over F4 — Digital
Digital (138, 162, 111797)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4162, 111797, F4, 2, 24) (dual of [(111797, 2), 223432, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4162, 131072, F4, 2, 24) (dual of [(131072, 2), 261982, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using
- OOA 2-folding [i] based on linear OA(4162, 262144, F4, 24) (dual of [262144, 261982, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(4162, 131072, F4, 2, 24) (dual of [(131072, 2), 261982, 25]-NRT-code), using
(162−24, 162, large)-Net in Base 4 — Upper bound on s
There is no (138, 162, large)-net in base 4, because
- 22 times m-reduction [i] would yield (138, 140, large)-net in base 4, but