Best Known (165, 165+37, s)-Nets in Base 2
(165, 165+37, 320)-Net over F2 — Constructive and digital
Digital (165, 202, 320)-net over F2, using
- 22 times duplication [i] based on digital (163, 200, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 40, 64)-net over F32, using
(165, 165+37, 750)-Net over F2 — Digital
Digital (165, 202, 750)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2202, 750, F2, 2, 37) (dual of [(750, 2), 1298, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2202, 1031, F2, 2, 37) (dual of [(1031, 2), 1860, 38]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2200, 1030, F2, 2, 37) (dual of [(1030, 2), 1860, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2200, 2060, F2, 37) (dual of [2060, 1860, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(2200, 2060, F2, 37) (dual of [2060, 1860, 38]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2200, 1030, F2, 2, 37) (dual of [(1030, 2), 1860, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2202, 1031, F2, 2, 37) (dual of [(1031, 2), 1860, 38]-NRT-code), using
(165, 165+37, 17336)-Net in Base 2 — Upper bound on s
There is no (165, 202, 17337)-net in base 2, because
- 1 times m-reduction [i] would yield (165, 201, 17337)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 214560 388612 789067 838359 989127 072795 390337 007082 232294 276400 > 2201 [i]