Best Known (113−14, 113, s)-Nets in Base 4
(113−14, 113, 599188)-Net over F4 — Constructive and digital
Digital (99, 113, 599188)-net over F4, using
- 41 times duplication [i] based on digital (98, 112, 599188)-net over F4, using
- net defined by OOA [i] based on linear OOA(4112, 599188, F4, 14, 14) (dual of [(599188, 14), 8388520, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4112, 4194316, F4, 14) (dual of [4194316, 4194204, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4111, 4194315, F4, 14) (dual of [4194315, 4194204, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4111, 4194315, F4, 14) (dual of [4194315, 4194204, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4112, 4194316, F4, 14) (dual of [4194316, 4194204, 15]-code), using
- net defined by OOA [i] based on linear OOA(4112, 599188, F4, 14, 14) (dual of [(599188, 14), 8388520, 15]-NRT-code), using
(113−14, 113, 1946362)-Net over F4 — Digital
Digital (99, 113, 1946362)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4113, 1946362, F4, 2, 14) (dual of [(1946362, 2), 3892611, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4113, 2097158, F4, 2, 14) (dual of [(2097158, 2), 4194203, 15]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4112, 2097158, F4, 2, 14) (dual of [(2097158, 2), 4194204, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4112, 4194316, F4, 14) (dual of [4194316, 4194204, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4111, 4194315, F4, 14) (dual of [4194315, 4194204, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4111, 4194315, F4, 14) (dual of [4194315, 4194204, 15]-code), using
- OOA 2-folding [i] based on linear OA(4112, 4194316, F4, 14) (dual of [4194316, 4194204, 15]-code), using
- 41 times duplication [i] based on linear OOA(4112, 2097158, F4, 2, 14) (dual of [(2097158, 2), 4194204, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4113, 2097158, F4, 2, 14) (dual of [(2097158, 2), 4194203, 15]-NRT-code), using
(113−14, 113, large)-Net in Base 4 — Upper bound on s
There is no (99, 113, large)-net in base 4, because
- 12 times m-reduction [i] would yield (99, 101, large)-net in base 4, but