Best Known (165−15, 165, s)-Nets in Base 4
(165−15, 165, 1199740)-Net over F4 — Constructive and digital
Digital (150, 165, 1199740)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (25, 32, 1369)-net over F4, using
- net defined by OOA [i] based on linear OOA(432, 1369, F4, 7, 7) (dual of [(1369, 7), 9551, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(432, 4108, F4, 7) (dual of [4108, 4076, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(432, 4109, F4, 7) (dual of [4109, 4077, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(432, 4109, F4, 7) (dual of [4109, 4077, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(432, 4108, F4, 7) (dual of [4108, 4076, 8]-code), using
- net defined by OOA [i] based on linear OOA(432, 1369, F4, 7, 7) (dual of [(1369, 7), 9551, 8]-NRT-code), using
- digital (118, 133, 1198371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(4133, large, F4, 15) (dual of [large, large−133, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4133, 8388598, F4, 15) (dual of [8388598, 8388465, 16]-code), using
- net defined by OOA [i] based on linear OOA(4133, 1198371, F4, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
- digital (25, 32, 1369)-net over F4, using
(165−15, 165, large)-Net over F4 — Digital
Digital (150, 165, large)-net over F4, using
- t-expansion [i] based on digital (149, 165, large)-net over F4, using
- 1 times m-reduction [i] based on digital (149, 166, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4166, large, F4, 17) (dual of [large, large−166, 18]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 21 times code embedding in larger space [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4166, large, F4, 17) (dual of [large, large−166, 18]-code), using
- 1 times m-reduction [i] based on digital (149, 166, large)-net over F4, using
(165−15, 165, large)-Net in Base 4 — Upper bound on s
There is no (150, 165, large)-net in base 4, because
- 13 times m-reduction [i] would yield (150, 152, large)-net in base 4, but