Best Known (76−19, 76, s)-Nets in Base 5
(76−19, 76, 347)-Net over F5 — Constructive and digital
Digital (57, 76, 347)-net over F5, using
- net defined by OOA [i] based on linear OOA(576, 347, F5, 19, 19) (dual of [(347, 19), 6517, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(576, 3124, F5, 19) (dual of [3124, 3048, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(576, 3124, F5, 19) (dual of [3124, 3048, 20]-code), using
(76−19, 76, 2164)-Net over F5 — Digital
Digital (57, 76, 2164)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(576, 2164, F5, 19) (dual of [2164, 2088, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using
(76−19, 76, 692528)-Net in Base 5 — Upper bound on s
There is no (57, 76, 692529)-net in base 5, because
- 1 times m-reduction [i] would yield (57, 75, 692529)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 26469 994051 278332 266843 443343 916065 725562 608924 748325 > 575 [i]