Best Known (89, 89+14, s)-Nets in Base 4
(89, 89+14, 149798)-Net over F4 — Constructive and digital
Digital (89, 103, 149798)-net over F4, using
- 42 times duplication [i] based on digital (87, 101, 149798)-net over F4, using
- net defined by OOA [i] based on linear OOA(4101, 149798, F4, 14, 14) (dual of [(149798, 14), 2097071, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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
- OA 7-folding and stacking [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- net defined by OOA [i] based on linear OOA(4101, 149798, F4, 14, 14) (dual of [(149798, 14), 2097071, 15]-NRT-code), using
(89, 89+14, 524294)-Net over F4 — Digital
Digital (89, 103, 524294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4103, 524294, F4, 2, 14) (dual of [(524294, 2), 1048485, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4101, 524293, F4, 2, 14) (dual of [(524293, 2), 1048485, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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
- OOA 2-folding [i] based on linear OA(4101, 1048586, F4, 14) (dual of [1048586, 1048485, 15]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4101, 524293, F4, 2, 14) (dual of [(524293, 2), 1048485, 15]-NRT-code), using
(89, 89+14, large)-Net in Base 4 — Upper bound on s
There is no (89, 103, large)-net in base 4, because
- 12 times m-reduction [i] would yield (89, 91, large)-net in base 4, but