Best Known (77−14, 77, s)-Nets in Base 4
(77−14, 77, 2344)-Net over F4 — Constructive and digital
Digital (63, 77, 2344)-net over F4, using
- 41 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(476, 2344, F4, 14, 14) (dual of [(2344, 14), 32740, 15]-NRT-code), using
(77−14, 77, 11454)-Net over F4 — Digital
Digital (63, 77, 11454)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(477, 11454, F4, 14) (dual of [11454, 11377, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(477, 16412, F4, 14) (dual of [16412, 16335, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(8) [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(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(477, 16412, F4, 14) (dual of [16412, 16335, 15]-code), using
(77−14, 77, 4725598)-Net in Base 4 — Upper bound on s
There is no (63, 77, 4725599)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 22835 984483 740482 562453 122639 407105 023168 694380 > 477 [i]