Best Known (33, 33+17, s)-Nets in Base 25
(33, 33+17, 1954)-Net over F25 — Constructive and digital
Digital (33, 50, 1954)-net over F25, using
- net defined by OOA [i] based on linear OOA(2550, 1954, F25, 17, 17) (dual of [(1954, 17), 33168, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2550, 15633, F25, 17) (dual of [15633, 15583, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(2549, 15626, F25, 17) (dual of [15626, 15577, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2543, 15626, F25, 15) (dual of [15626, 15583, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,8]) ⊂ C([0,7]) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(2550, 15633, F25, 17) (dual of [15633, 15583, 18]-code), using
(33, 33+17, 9859)-Net over F25 — Digital
Digital (33, 50, 9859)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2550, 9859, F25, 17) (dual of [9859, 9809, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2550, 15633, F25, 17) (dual of [15633, 15583, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(2549, 15626, F25, 17) (dual of [15626, 15577, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2543, 15626, F25, 15) (dual of [15626, 15583, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2550, 15633, F25, 17) (dual of [15633, 15583, 18]-code), using
(33, 33+17, large)-Net in Base 25 — Upper bound on s
There is no (33, 50, large)-net in base 25, because
- 15 times m-reduction [i] would yield (33, 35, large)-net in base 25, but