Best Known (70−35, 70, s)-Nets in Base 25
(70−35, 70, 208)-Net over F25 — Constructive and digital
Digital (35, 70, 208)-net over F25, using
- net from sequence [i] based on digital (35, 207)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 35 and N(F) ≥ 208, using
(70−35, 70, 445)-Net over F25 — Digital
Digital (35, 70, 445)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2570, 445, F25, 35) (dual of [445, 375, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2570, 631, F25, 35) (dual of [631, 561, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [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(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(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2570, 631, F25, 35) (dual of [631, 561, 36]-code), using
(70−35, 70, 141155)-Net in Base 25 — Upper bound on s
There is no (35, 70, 141156)-net in base 25, because
- 1 times m-reduction [i] would yield (35, 69, 141156)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2 869900 797939 583684 394003 368639 805326 103795 875801 680123 828338 425463 582107 133320 477641 251139 165025 > 2569 [i]