Best Known (86, 86+13, s)-Nets in Base 25
(86, 86+13, 2804050)-Net over F25 — Constructive and digital
Digital (86, 99, 2804050)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (8, 12, 7850)-net over F25, using
- net defined by OOA [i] based on linear OOA(2512, 7850, F25, 4, 4) (dual of [(7850, 4), 31388, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2512, 15700, F25, 4) (dual of [15700, 15688, 5]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(250, 628, F25, 0) (dual of [628, 628, 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
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code) (see above)
- linear OA(251, 628, F25, 1) (dual of [628, 627, 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
- linear OA(251, 628, F25, 1) (dual of [628, 627, 2]-code) (see above)
- linear OA(253, 628, F25, 2) (dual of [628, 625, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 651, F25, 2) (dual of [651, 648, 3]-code), using
- Hamming code H(3,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 651, F25, 2) (dual of [651, 648, 3]-code), using
- linear OA(257, 628, F25, 4) (dual of [628, 621, 5]-code), using
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(255, 624, F25, 3) (dual of [624, 619, 4]-code or 624-cap in PG(4,25)), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(257, 624, F25, 4) (dual of [624, 617, 5]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(253, 624, F25, 2) (dual of [624, 621, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 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 (see above)
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([623,2]) [i] based on
- linear OA(250, 628, F25, 0) (dual of [628, 628, 1]-code), using
- generalized (u, u+v)-construction [i] based on
- OA 2-folding and stacking [i] based on linear OA(2512, 15700, F25, 4) (dual of [15700, 15688, 5]-code), using
- net defined by OOA [i] based on linear OOA(2512, 7850, F25, 4, 4) (dual of [(7850, 4), 31388, 5]-NRT-code), using
- digital (20, 26, 1398100)-net over F25, using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- digital (48, 61, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (8, 12, 7850)-net over F25, using
(86, 86+13, large)-Net over F25 — Digital
Digital (86, 99, large)-net over F25, using
- t-expansion [i] based on digital (85, 99, large)-net over F25, using
- 8 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
- 8 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
(86, 86+13, large)-Net in Base 25 — Upper bound on s
There is no (86, 99, large)-net in base 25, because
- 11 times m-reduction [i] would yield (86, 88, large)-net in base 25, but