Best Known (213, 213+15, s)-Nets in Base 4
(213, 213+15, 6191588)-Net over F4 — Constructive and digital
Digital (213, 228, 6191588)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (49, 56, 1398104)-net over F4, using
- net defined by OOA [i] based on linear OOA(456, 1398104, F4, 7, 7) (dual of [(1398104, 7), 9786672, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(456, 4194313, F4, 7) (dual of [4194313, 4194257, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(456, 4194315, F4, 7) (dual of [4194315, 4194259, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(456, 4194304, F4, 7) (dual of [4194304, 4194248, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(456, 4194315, F4, 7) (dual of [4194315, 4194259, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(456, 4194313, F4, 7) (dual of [4194313, 4194257, 8]-code), using
- net defined by OOA [i] based on linear OOA(456, 1398104, F4, 7, 7) (dual of [(1398104, 7), 9786672, 8]-NRT-code), using
- digital (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (49, 56, 1398104)-net over F4, using
(213, 213+15, large)-Net over F4 — Digital
Digital (213, 228, large)-net over F4, using
- 9 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
(213, 213+15, large)-Net in Base 4 — Upper bound on s
There is no (213, 228, large)-net in base 4, because
- 13 times m-reduction [i] would yield (213, 215, large)-net in base 4, but