Best Known (53, 81, s)-Nets in Base 25
(53, 81, 1116)-Net over F25 — Constructive and digital
Digital (53, 81, 1116)-net over F25, using
- 1 times m-reduction [i] based on digital (53, 82, 1116)-net over F25, using
- net defined by OOA [i] based on linear OOA(2582, 1116, F25, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 14-folding and stacking with additional row [i] based on linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using
- net defined by OOA [i] based on linear OOA(2582, 1116, F25, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
(53, 81, 8786)-Net over F25 — Digital
Digital (53, 81, 8786)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2581, 8786, F25, 28) (dual of [8786, 8705, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, 15633, F25, 28) (dual of [15633, 15552, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2580, 15632, F25, 28) (dual of [15632, 15552, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(2579, 15625, F25, 28) (dual of [15625, 15546, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2573, 15625, F25, 26) (dual of [15625, 15552, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(27) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2580, 15632, F25, 28) (dual of [15632, 15552, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, 15633, F25, 28) (dual of [15633, 15552, 29]-code), using
(53, 81, large)-Net in Base 25 — Upper bound on s
There is no (53, 81, large)-net in base 25, because
- 26 times m-reduction [i] would yield (53, 55, large)-net in base 25, but