Best Known (76−14, 76, s)-Nets in Base 4
(76−14, 76, 2344)-Net over F4 — Constructive and digital
Digital (62, 76, 2344)-net over F4, using
- net defined by OOA [i] based on linear OOA(476, 2344, F4, 14, 14) (dual of [(2344, 14), 32740, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(476, 16408, F4, 14) (dual of [16408, 16332, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 16410, F4, 14) (dual of [16410, 16334, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 16410, F4, 14) (dual of [16410, 16334, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(476, 16408, F4, 14) (dual of [16408, 16332, 15]-code), using
(76−14, 76, 10203)-Net over F4 — Digital
Digital (62, 76, 10203)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(476, 10203, F4, 14) (dual of [10203, 10127, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 16410, F4, 14) (dual of [16410, 16334, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 16410, F4, 14) (dual of [16410, 16334, 15]-code), using
(76−14, 76, 3876574)-Net in Base 4 — Upper bound on s
There is no (62, 76, 3876575)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5708 996329 133524 689604 808297 881690 588742 399596 > 476 [i]