Best Known (83−29, 83, s)-Nets in Base 25
(83−29, 83, 1116)-Net over F25 — Constructive and digital
Digital (54, 83, 1116)-net over F25, using
- 251 times duplication [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
(83−29, 83, 7999)-Net over F25 — Digital
Digital (54, 83, 7999)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2583, 7999, F25, 29) (dual of [7999, 7916, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2583, 15632, F25, 29) (dual of [15632, 15549, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [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(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(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(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2583, 15632, F25, 29) (dual of [15632, 15549, 30]-code), using
(83−29, 83, large)-Net in Base 25 — Upper bound on s
There is no (54, 83, large)-net in base 25, because
- 27 times m-reduction [i] would yield (54, 56, large)-net in base 25, but