Best Known (61, 61+13, s)-Nets in Base 25
(61, 61+13, 1398311)-Net over F25 — Constructive and digital
Digital (61, 74, 1398311)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 211)-net over F25, using
- net defined by OOA [i] based on linear OOA(2513, 211, F25, 6, 6) (dual of [(211, 6), 1253, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2513, 633, F25, 6) (dual of [633, 620, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(2511, 625, F25, 6) (dual of [625, 614, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(255, 625, F25, 3) (dual of [625, 620, 4]-code or 625-cap in PG(4,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2513, 633, F25, 6) (dual of [633, 620, 7]-code), using
- net defined by OOA [i] based on linear OOA(2513, 211, F25, 6, 6) (dual of [(211, 6), 1253, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 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 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (7, 13, 211)-net over F25, using
(61, 61+13, large)-Net over F25 — Digital
Digital (61, 74, large)-net over F25, using
- 3 times m-reduction [i] based on digital (61, 77, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2577, large, F25, 16) (dual of [large, large−77, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 1 times code embedding in larger space [i] based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2577, large, F25, 16) (dual of [large, large−77, 17]-code), using
(61, 61+13, large)-Net in Base 25 — Upper bound on s
There is no (61, 74, large)-net in base 25, because
- 11 times m-reduction [i] would yield (61, 63, large)-net in base 25, but