Best Known (88−13, 88, s)-Nets in Base 4
(88−13, 88, 43696)-Net over F4 — Constructive and digital
Digital (75, 88, 43696)-net over F4, using
- net defined by OOA [i] based on linear OOA(488, 43696, F4, 13, 13) (dual of [(43696, 13), 567960, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(488, 262177, F4, 13) (dual of [262177, 262089, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(488, 262177, F4, 13) (dual of [262177, 262089, 14]-code), using
(88−13, 88, 131088)-Net over F4 — Digital
Digital (75, 88, 131088)-net over F4, using
- 41 times duplication [i] based on digital (74, 87, 131088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(487, 131088, F4, 2, 13) (dual of [(131088, 2), 262089, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(487, 262176, F4, 13) (dual of [262176, 262089, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(487, 131088, F4, 2, 13) (dual of [(131088, 2), 262089, 14]-NRT-code), using
(88−13, 88, large)-Net in Base 4 — Upper bound on s
There is no (75, 88, large)-net in base 4, because
- 11 times m-reduction [i] would yield (75, 77, large)-net in base 4, but