Best Known (11, 11+9, s)-Nets in Base 25
(11, 11+9, 159)-Net over F25 — Constructive and digital
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
(11, 11+9, 670)-Net over F25 — Digital
Digital (11, 20, 670)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2520, 670, F25, 9) (dual of [670, 650, 10]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0) [i] based on linear OA(2517, 628, F25, 9) (dual of [628, 611, 10]-code), using
- construction XX applied to C1 = C([623,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([623,7]) [i] based on
- 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(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2517, 624, F25, 9) (dual of [624, 607, 10]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [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(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([623,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([623,7]) [i] based on
- 39 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0) [i] based on linear OA(2517, 628, F25, 9) (dual of [628, 611, 10]-code), using
(11, 11+9, 402767)-Net in Base 25 — Upper bound on s
There is no (11, 20, 402768)-net in base 25, because
- 1 times m-reduction [i] would yield (11, 19, 402768)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 363 798368 827934 846525 397505 > 2519 [i]