Best Known (53, 53+29, s)-Nets in Base 25
(53, 53+29, 1116)-Net over F25 — Constructive and digital
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
(53, 53+29, 7814)-Net over F25 — Digital
Digital (53, 82, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2582, 7814, F25, 2, 29) (dual of [(7814, 2), 15546, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2582, 15628, F25, 29) (dual of [15628, 15546, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [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]
- 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(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(2582, 15628, F25, 29) (dual of [15628, 15546, 30]-code), using
(53, 53+29, large)-Net in Base 25 — Upper bound on s
There is no (53, 82, large)-net in base 25, because
- 27 times m-reduction [i] would yield (53, 55, large)-net in base 25, but