Best Known (57, 57+18, s)-Nets in Base 25
(57, 57+18, 43406)-Net over F25 — Constructive and digital
Digital (57, 75, 43406)-net over F25, using
- net defined by OOA [i] based on linear OOA(2575, 43406, F25, 18, 18) (dual of [(43406, 18), 781233, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2575, 390654, F25, 18) (dual of [390654, 390579, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2545, 390625, F25, 12) (dual of [390625, 390580, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(256, 29, F25, 5) (dual of [29, 23, 6]-code), using
- construction for [s, s−6, 6]-code from [s, 3, s−3]-code [i] based on linear OA(2526, 29, F25, 25) (dual of [29, 3, 26]-code), using
- construction X applied to C1 ⊂ C0 [i] based on
- linear OA(2525, 26, F25, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,25)), using code C1 for u = 2 by de Boer and Brouwer [i]
- linear OA(2523, 26, F25, 23) (dual of [26, 3, 24]-code or 26-arc in PG(22,25)), using code C0 for u = 2 by de Boer and Brouwer [i]
- linear OA(251, 3, F25, 1) (dual of [3, 2, 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 C1 ⊂ C0 [i] based on
- construction for [s, s−6, 6]-code from [s, 3, s−3]-code [i] based on linear OA(2526, 29, F25, 25) (dual of [29, 3, 26]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- OA 9-folding and stacking [i] based on linear OA(2575, 390654, F25, 18) (dual of [390654, 390579, 19]-code), using
(57, 57+18, 439668)-Net over F25 — Digital
Digital (57, 75, 439668)-net over F25, using
(57, 57+18, large)-Net in Base 25 — Upper bound on s
There is no (57, 75, large)-net in base 25, because
- 16 times m-reduction [i] would yield (57, 59, large)-net in base 25, but