Best Known (106−18, 106, s)-Nets in Base 25
(106−18, 106, 932226)-Net over F25 — Constructive and digital
Digital (88, 106, 932226)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (11, 20, 159)-net over F25, using
- net defined by OOA [i] based on linear OOA(2520, 159, F25, 9, 9) (dual of [(159, 9), 1411, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2520, 637, F25, 9) (dual of [637, 617, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(2517, 626, F25, 9) (dual of [626, 609, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(259, 626, F25, 5) (dual of [626, 617, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(253, 11, F25, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,25) or 11-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(2520, 637, F25, 9) (dual of [637, 617, 10]-code), using
- net defined by OOA [i] based on linear OOA(2520, 159, F25, 9, 9) (dual of [(159, 9), 1411, 10]-NRT-code), using
- digital (68, 86, 932067)-net over F25, using
- net defined by OOA [i] based on linear OOA(2586, 932067, F25, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- net defined by OOA [i] based on linear OOA(2586, 932067, F25, 18, 18) (dual of [(932067, 18), 16777120, 19]-NRT-code), using
- digital (11, 20, 159)-net over F25, using
(106−18, 106, large)-Net over F25 — Digital
Digital (88, 106, large)-net over F25, using
- t-expansion [i] based on digital (85, 106, large)-net over F25, using
- 1 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 1 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
(106−18, 106, large)-Net in Base 25 — Upper bound on s
There is no (88, 106, large)-net in base 25, because
- 16 times m-reduction [i] would yield (88, 90, large)-net in base 25, but