Best Known (83, 83+16, s)-Nets in Base 25
(83, 83+16, 1052483)-Net over F25 — Constructive and digital
Digital (83, 99, 1052483)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (15, 23, 3908)-net over F25, using
- net defined by OOA [i] based on linear OOA(2523, 3908, F25, 8, 8) (dual of [(3908, 8), 31241, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2523, 15632, F25, 8) (dual of [15632, 15609, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(2522, 15625, F25, 8) (dual of [15625, 15603, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2516, 15625, F25, 6) (dual of [15625, 15609, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2523, 15632, F25, 8) (dual of [15632, 15609, 9]-code), using
- net defined by OOA [i] based on linear OOA(2523, 3908, F25, 8, 8) (dual of [(3908, 8), 31241, 9]-NRT-code), using
- digital (60, 76, 1048575)-net over F25, using
- net defined by OOA [i] based on linear OOA(2576, 1048575, F25, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2576, 8388600, F25, 16) (dual of [8388600, 8388524, 17]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2576, 8388600, F25, 16) (dual of [8388600, 8388524, 17]-code), using
- net defined by OOA [i] based on linear OOA(2576, 1048575, F25, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- digital (15, 23, 3908)-net over F25, using
(83, 83+16, large)-Net over F25 — Digital
Digital (83, 99, large)-net over F25, using
- t-expansion [i] based on digital (81, 99, large)-net over F25, using
- 3 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 3 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
(83, 83+16, large)-Net in Base 25 — Upper bound on s
There is no (83, 99, large)-net in base 25, because
- 14 times m-reduction [i] would yield (83, 85, large)-net in base 25, but