Best Known (54−25, 54, s)-Nets in Base 25
(54−25, 54, 200)-Net over F25 — Constructive and digital
Digital (29, 54, 200)-net over F25, using
- t-expansion [i] based on digital (25, 54, 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
(54−25, 54, 643)-Net over F25 — Digital
Digital (29, 54, 643)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2554, 643, F25, 25) (dual of [643, 589, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(2549, 626, F25, 25) (dual of [626, 577, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2537, 626, F25, 19) (dual of [626, 589, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(255, 17, F25, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- Reed–Solomon code RS(20,25) [i]
- discarding factors / shortening the dual code based on linear OA(255, 25, F25, 5) (dual of [25, 20, 6]-code or 25-arc in PG(4,25)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
(54−25, 54, 329137)-Net in Base 25 — Upper bound on s
There is no (29, 54, 329138)-net in base 25, because
- 1 times m-reduction [i] would yield (29, 53, 329138)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 123 262010 872201 869024 565836 584279 352462 632224 387621 064079 377887 254575 870401 > 2553 [i]