Best Known (46−8, 46, s)-Nets in Base 4
(46−8, 46, 4101)-Net over F4 — Constructive and digital
Digital (38, 46, 4101)-net over F4, using
- net defined by OOA [i] based on linear OOA(446, 4101, F4, 8, 8) (dual of [(4101, 8), 32762, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(446, 16404, F4, 8) (dual of [16404, 16358, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(446, 16405, F4, 8) (dual of [16405, 16359, 9]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(446, 16405, F4, 8) (dual of [16405, 16359, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(446, 16404, F4, 8) (dual of [16404, 16358, 9]-code), using
(46−8, 46, 16405)-Net over F4 — Digital
Digital (38, 46, 16405)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(446, 16405, F4, 8) (dual of [16405, 16359, 9]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
(46−8, 46, 6189011)-Net in Base 4 — Upper bound on s
There is no (38, 46, 6189012)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4951 763240 034962 258325 624484 > 446 [i]