Best Known (62−29, 62, s)-Nets in Base 25
(62−29, 62, 230)-Net over F25 — Constructive and digital
Digital (33, 62, 230)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 23, 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, 39, 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, 23, 104)-net over F25, using
(62−29, 62, 643)-Net over F25 — Digital
Digital (33, 62, 643)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2562, 643, F25, 29) (dual of [643, 581, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,11]) [i] based on
- 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(2545, 626, F25, 23) (dual of [626, 581, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,14]) ⊂ C([0,11]) [i] based on
(62−29, 62, 310642)-Net in Base 25 — Upper bound on s
There is no (33, 62, 310643)-net in base 25, because
- 1 times m-reduction [i] would yield (33, 61, 310643)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 18 808305 634266 285670 265227 493404 886563 223498 287848 335736 014603 303291 339805 974174 120305 > 2561 [i]