Best Known (69, 69+11, s)-Nets in Base 25
(69, 69+11, 4314191)-Net over F25 — Constructive and digital
Digital (69, 80, 4314191)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 8, 958751)-net over F25, using
- net defined by OOA [i] based on linear OOA(258, 958751, F25, 3, 3) (dual of [(958751, 3), 2876245, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(258, 958751, F25, 2, 3) (dual of [(958751, 2), 1917494, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(258, 958751, F25, 3) (dual of [958751, 958743, 4]-code or 958751-cap in PG(7,25)), using
- appending kth column [i] based on linear OOA(258, 958751, F25, 2, 3) (dual of [(958751, 2), 1917494, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(258, 958751, F25, 3, 3) (dual of [(958751, 3), 2876245, 4]-NRT-code), using
- digital (16, 21, 1677720)-net over F25, using
- s-reduction based on digital (16, 21, 4194301)-net over F25, using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- s-reduction based on digital (16, 21, 4194301)-net over F25, using
- digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (5, 8, 958751)-net over F25, using
(69, 69+11, large)-Net over F25 — Digital
Digital (69, 80, large)-net over F25, using
- 7 times m-reduction [i] based on digital (69, 87, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
(69, 69+11, large)-Net in Base 25 — Upper bound on s
There is no (69, 80, large)-net in base 25, because
- 9 times m-reduction [i] would yield (69, 71, large)-net in base 25, but