Best Known (26, 51, s)-Nets in Base 25
(26, 51, 200)-Net over F25 — Constructive and digital
Digital (26, 51, 200)-net over F25, using
- t-expansion [i] based on digital (25, 51, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
(26, 51, 419)-Net over F25 — Digital
Digital (26, 51, 419)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2551, 419, F25, 25) (dual of [419, 368, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, 634, F25, 25) (dual of [634, 583, 26]-code), using
- construction XX applied to C1 = C([621,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([621,21]) [i] based on
- linear OA(2547, 624, F25, 24) (dual of [624, 577, 25]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−3,−2,…,20}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2543, 624, F25, 22) (dual of [624, 581, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2549, 624, F25, 25) (dual of [624, 575, 26]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−3,−2,…,21}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2541, 624, F25, 21) (dual of [624, 583, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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
- 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
- construction XX applied to C1 = C([621,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([621,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2551, 634, F25, 25) (dual of [634, 583, 26]-code), using
(26, 51, 147191)-Net in Base 25 — Upper bound on s
There is no (26, 51, 147192)-net in base 25, because
- 1 times m-reduction [i] would yield (26, 50, 147192)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 7888 979817 297655 190376 516542 014742 940036 868242 878234 702580 016549 311745 > 2550 [i]