Best Known (39−22, 39, s)-Nets in Base 81
(39−22, 39, 370)-Net over F81 — Constructive and digital
Digital (17, 39, 370)-net over F81, using
- t-expansion [i] based on digital (16, 39, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(39−22, 39, 412)-Net over F81 — Digital
Digital (17, 39, 412)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8139, 412, F81, 22) (dual of [412, 373, 23]-code), using
- construction X applied to C([30,51]) ⊂ C([31,51]) [i] based on
- linear OA(8139, 410, F81, 22) (dual of [410, 371, 23]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {30,31,…,51}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8137, 410, F81, 21) (dual of [410, 373, 22]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {31,32,…,51}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([30,51]) ⊂ C([31,51]) [i] based on
(39−22, 39, 358397)-Net in Base 81 — Upper bound on s
There is no (17, 39, 358398)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 269 726719 386596 244861 286291 292594 839726 903748 736441 598112 418205 474312 542241 > 8139 [i]