Best Known (9, 17, s)-Nets in Base 25
(9, 17, 158)-Net over F25 — Constructive and digital
Digital (9, 17, 158)-net over F25, using
- net defined by OOA [i] based on linear OOA(2517, 158, F25, 8, 8) (dual of [(158, 8), 1247, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2517, 632, F25, 8) (dual of [632, 615, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2517, 633, F25, 8) (dual of [633, 616, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(2515, 625, F25, 8) (dual of [625, 610, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(259, 625, F25, 5) (dual of [625, 616, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(2517, 633, F25, 8) (dual of [633, 616, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2517, 632, F25, 8) (dual of [632, 615, 9]-code), using
(9, 17, 641)-Net over F25 — Digital
Digital (9, 17, 641)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2517, 641, F25, 8) (dual of [641, 624, 9]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0) [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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, using
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- 11 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0) [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
(9, 17, 80552)-Net in Base 25 — Upper bound on s
There is no (9, 17, 80553)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 582098 586181 643733 184225 > 2517 [i]