Best Known (79−28, 79, s)-Nets in Base 25
(79−28, 79, 1116)-Net over F25 — Constructive and digital
Digital (51, 79, 1116)-net over F25, using
- net defined by OOA [i] based on linear OOA(2579, 1116, F25, 28, 28) (dual of [(1116, 28), 31169, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2579, 15624, F25, 28) (dual of [15624, 15545, 29]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2579, 15625, F25, 28) (dual of [15625, 15546, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2579, 15624, F25, 28) (dual of [15624, 15545, 29]-code), using
(79−28, 79, 7814)-Net over F25 — Digital
Digital (51, 79, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2579, 7814, F25, 2, 28) (dual of [(7814, 2), 15549, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2579, 15628, F25, 28) (dual of [15628, 15549, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [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(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(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(27) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(2579, 15628, F25, 28) (dual of [15628, 15549, 29]-code), using
(79−28, 79, large)-Net in Base 25 — Upper bound on s
There is no (51, 79, large)-net in base 25, because
- 26 times m-reduction [i] would yield (51, 53, large)-net in base 25, but