Best Known (101−13, 101, s)-Nets in Base 4
(101−13, 101, 699052)-Net over F4 — Constructive and digital
Digital (88, 101, 699052)-net over F4, using
- net defined by OOA [i] based on linear OOA(4101, 699052, F4, 13, 13) (dual of [(699052, 13), 9087575, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4101, 4194313, F4, 13) (dual of [4194313, 4194212, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 4194316, F4, 13) (dual of [4194316, 4194215, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [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(489, 4194304, F4, 11) (dual of [4194304, 4194215, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4101, 4194316, F4, 13) (dual of [4194316, 4194215, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4101, 4194313, F4, 13) (dual of [4194313, 4194212, 14]-code), using
(101−13, 101, 1398105)-Net over F4 — Digital
Digital (88, 101, 1398105)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4101, 1398105, F4, 3, 13) (dual of [(1398105, 3), 4194214, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4101, 4194315, F4, 13) (dual of [4194315, 4194214, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 4194316, F4, 13) (dual of [4194316, 4194215, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [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(489, 4194304, F4, 11) (dual of [4194304, 4194215, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4101, 4194316, F4, 13) (dual of [4194316, 4194215, 14]-code), using
- OOA 3-folding [i] based on linear OA(4101, 4194315, F4, 13) (dual of [4194315, 4194214, 14]-code), using
(101−13, 101, large)-Net in Base 4 — Upper bound on s
There is no (88, 101, large)-net in base 4, because
- 11 times m-reduction [i] would yield (88, 90, large)-net in base 4, but