Best Known (77, 77+13, s)-Nets in Base 25
(77, 77+13, 2796200)-Net over F25 — Constructive and digital
Digital (77, 90, 2796200)-net over F25, using
- 253 times duplication [i] based on digital (74, 87, 2796200)-net over F25, using
- net defined by OOA [i] based on linear OOA(2587, 2796200, F25, 14, 13) (dual of [(2796200, 14), 39146713, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2587, 8388601, F25, 2, 13) (dual of [(8388601, 2), 16777115, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2587, 8388602, F25, 2, 13) (dual of [(8388602, 2), 16777117, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2526, 4194301, F25, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2526, 8388602, F25, 6) (dual of [8388602, 8388576, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- OOA 2-folding [i] based on linear OA(2526, 8388602, F25, 6) (dual of [8388602, 8388576, 7]-code), using
- linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- linear OOA(2526, 4194301, F25, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2587, 8388602, F25, 2, 13) (dual of [(8388602, 2), 16777117, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(2587, 8388601, F25, 2, 13) (dual of [(8388601, 2), 16777115, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2587, 2796200, F25, 14, 13) (dual of [(2796200, 14), 39146713, 14]-NRT-code), using
(77, 77+13, large)-Net over F25 — Digital
Digital (77, 90, large)-net over F25, using
- 7 times m-reduction [i] based on digital (77, 97, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
(77, 77+13, large)-Net in Base 25 — Upper bound on s
There is no (77, 90, large)-net in base 25, because
- 11 times m-reduction [i] would yield (77, 79, large)-net in base 25, but