Best Known (72−35, 72, s)-Nets in Base 25
(72−35, 72, 252)-Net over F25 — Constructive and digital
Digital (37, 72, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 27, 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 (10, 45, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 27, 126)-net over F25, using
(72−35, 72, 543)-Net over F25 — Digital
Digital (37, 72, 543)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2572, 543, F25, 35) (dual of [543, 471, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2572, 637, F25, 35) (dual of [637, 565, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,15]) [i] based on
- linear OA(2569, 626, F25, 35) (dual of [626, 557, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(2561, 626, F25, 31) (dual of [626, 565, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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([0,17]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2572, 637, F25, 35) (dual of [637, 565, 36]-code), using
(72−35, 72, 206144)-Net in Base 25 — Upper bound on s
There is no (37, 72, 206145)-net in base 25, because
- 1 times m-reduction [i] would yield (37, 71, 206145)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1793 792336 696007 357274 308947 076286 656768 376544 374304 847771 649636 464221 952038 026301 982408 483319 480729 > 2571 [i]