Best Known (133−17, 133, s)-Nets in Base 4
(133−17, 133, 524288)-Net over F4 — Constructive and digital
Digital (116, 133, 524288)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 524288, F4, 17, 17) (dual of [(524288, 17), 8912763, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4133, 4194305, F4, 17) (dual of [4194305, 4194172, 18]-code), using
(133−17, 133, 1398101)-Net over F4 — Digital
Digital (116, 133, 1398101)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4133, 1398101, F4, 3, 17) (dual of [(1398101, 3), 4194170, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4133, 4194303, F4, 17) (dual of [4194303, 4194170, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using
- OOA 3-folding [i] based on linear OA(4133, 4194303, F4, 17) (dual of [4194303, 4194170, 18]-code), using
(133−17, 133, large)-Net in Base 4 — Upper bound on s
There is no (116, 133, large)-net in base 4, because
- 15 times m-reduction [i] would yield (116, 118, large)-net in base 4, but