Best Known (35, 35+33, s)-Nets in Base 25
(35, 35+33, 230)-Net over F25 — Constructive and digital
Digital (35, 68, 230)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 25, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (10, 43, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (9, 25, 104)-net over F25, using
(35, 35+33, 529)-Net over F25 — Digital
Digital (35, 68, 529)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2568, 529, F25, 33) (dual of [529, 461, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 637, F25, 33) (dual of [637, 569, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2565, 626, F25, 33) (dual of [626, 561, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2557, 626, F25, 29) (dual of [626, 569, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2568, 637, F25, 33) (dual of [637, 569, 34]-code), using
(35, 35+33, 202387)-Net in Base 25 — Upper bound on s
There is no (35, 68, 202388)-net in base 25, because
- 1 times m-reduction [i] would yield (35, 67, 202388)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 4591 946250 923444 417538 870008 390412 020793 416618 547069 213398 519773 041965 633124 776420 379376 065025 > 2567 [i]