Best Known (77−27, 77, s)-Nets in Base 25
(77−27, 77, 1202)-Net over F25 — Constructive and digital
Digital (50, 77, 1202)-net over F25, using
- 251 times duplication [i] based on digital (49, 76, 1202)-net over F25, using
- net defined by OOA [i] based on linear OOA(2576, 1202, F25, 27, 27) (dual of [(1202, 27), 32378, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2576, 15627, F25, 27) (dual of [15627, 15551, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2576, 15628, F25, 27) (dual of [15628, 15552, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(2576, 15625, F25, 27) (dual of [15625, 15549, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(2576, 15628, F25, 27) (dual of [15628, 15552, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2576, 15627, F25, 27) (dual of [15627, 15551, 28]-code), using
- net defined by OOA [i] based on linear OOA(2576, 1202, F25, 27, 27) (dual of [(1202, 27), 32378, 28]-NRT-code), using
(77−27, 77, 7814)-Net over F25 — Digital
Digital (50, 77, 7814)-net over F25, using
- 251 times duplication [i] based on digital (49, 76, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2576, 7814, F25, 2, 27) (dual of [(7814, 2), 15552, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2576, 15628, F25, 27) (dual of [15628, 15552, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(2576, 15625, F25, 27) (dual of [15625, 15549, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(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(26) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(2576, 15628, F25, 27) (dual of [15628, 15552, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2576, 7814, F25, 2, 27) (dual of [(7814, 2), 15552, 28]-NRT-code), using
(77−27, 77, large)-Net in Base 25 — Upper bound on s
There is no (50, 77, large)-net in base 25, because
- 25 times m-reduction [i] would yield (50, 52, large)-net in base 25, but