Best Known (58, 67, s)-Nets in Base 4
(58, 67, 1048576)-Net over F4 — Constructive and digital
Digital (58, 67, 1048576)-net over F4, using
- net defined by OOA [i] based on linear OOA(467, 1048576, F4, 9, 9) (dual of [(1048576, 9), 9437117, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
(58, 67, 2097152)-Net over F4 — Digital
Digital (58, 67, 2097152)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(467, 2097152, F4, 2, 9) (dual of [(2097152, 2), 4194237, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using
(58, 67, large)-Net in Base 4 — Upper bound on s
There is no (58, 67, large)-net in base 4, because
- 7 times m-reduction [i] would yield (58, 60, large)-net in base 4, but