Best Known (74, 91, s)-Nets in Base 4
(74, 91, 2051)-Net over F4 — Constructive and digital
Digital (74, 91, 2051)-net over F4, using
- 41 times duplication [i] based on digital (73, 90, 2051)-net over F4, using
- net defined by OOA [i] based on linear OOA(490, 2051, F4, 17, 17) (dual of [(2051, 17), 34777, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(490, 16409, F4, 17) (dual of [16409, 16319, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(490, 16409, F4, 17) (dual of [16409, 16319, 18]-code), using
- net defined by OOA [i] based on linear OOA(490, 2051, F4, 17, 17) (dual of [(2051, 17), 34777, 18]-NRT-code), using
(74, 91, 8759)-Net over F4 — Digital
Digital (74, 91, 8759)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(491, 8759, F4, 17) (dual of [8759, 8668, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 16411, F4, 17) (dual of [16411, 16320, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 16411, F4, 17) (dual of [16411, 16320, 18]-code), using
(74, 91, 7442920)-Net in Base 4 — Upper bound on s
There is no (74, 91, 7442921)-net in base 4, because
- 1 times m-reduction [i] would yield (74, 90, 7442921)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 532497 122884 123973 579139 114549 082178 824698 350132 064309 > 490 [i]