Best Known (106−13, 106, s)-Nets in Base 4
(106−13, 106, 699057)-Net over F4 — Constructive and digital
Digital (93, 106, 699057)-net over F4, using
- net defined by OOA [i] based on linear OOA(4106, 699057, F4, 13, 13) (dual of [(699057, 13), 9087635, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4106, 4194343, F4, 13) (dual of [4194343, 4194237, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4105, 4194342, F4, 13) (dual of [4194342, 4194237, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4105, 4194342, F4, 13) (dual of [4194342, 4194237, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4106, 4194343, F4, 13) (dual of [4194343, 4194237, 14]-code), using
(106−13, 106, 2097171)-Net over F4 — Digital
Digital (93, 106, 2097171)-net over F4, using
- 41 times duplication [i] based on digital (92, 105, 2097171)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4105, 2097171, F4, 2, 13) (dual of [(2097171, 2), 4194237, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4105, 4194342, F4, 13) (dual of [4194342, 4194237, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(4105, 4194342, F4, 13) (dual of [4194342, 4194237, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4105, 2097171, F4, 2, 13) (dual of [(2097171, 2), 4194237, 14]-NRT-code), using
(106−13, 106, large)-Net in Base 4 — Upper bound on s
There is no (93, 106, large)-net in base 4, because
- 11 times m-reduction [i] would yield (93, 95, large)-net in base 4, but