Best Known (88−31, 88, s)-Nets in Base 25
(88−31, 88, 1041)-Net over F25 — Constructive and digital
Digital (57, 88, 1041)-net over F25, using
- net defined by OOA [i] based on linear OOA(2588, 1041, F25, 31, 31) (dual of [(1041, 31), 32183, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2588, 15616, F25, 31) (dual of [15616, 15528, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2588, 15625, F25, 31) (dual of [15625, 15537, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(2588, 15625, F25, 31) (dual of [15625, 15537, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2588, 15616, F25, 31) (dual of [15616, 15528, 32]-code), using
(88−31, 88, 7814)-Net over F25 — Digital
Digital (57, 88, 7814)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2588, 7814, F25, 2, 31) (dual of [(7814, 2), 15540, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2588, 15628, F25, 31) (dual of [15628, 15540, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(2588, 15625, F25, 31) (dual of [15625, 15537, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2585, 15625, F25, 30) (dual of [15625, 15540, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(2588, 15628, F25, 31) (dual of [15628, 15540, 32]-code), using
(88−31, 88, large)-Net in Base 25 — Upper bound on s
There is no (57, 88, large)-net in base 25, because
- 29 times m-reduction [i] would yield (57, 59, large)-net in base 25, but