Best Known (239−37, 239, s)-Nets in Base 2
(239−37, 239, 520)-Net over F2 — Constructive and digital
Digital (202, 239, 520)-net over F2, using
- t-expansion [i] based on digital (201, 239, 520)-net over F2, using
- 1 times m-reduction [i] based on digital (201, 240, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 48, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 48, 104)-net over F32, using
- 1 times m-reduction [i] based on digital (201, 240, 520)-net over F2, using
(239−37, 239, 2031)-Net over F2 — Digital
Digital (202, 239, 2031)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 2031, F2, 4, 37) (dual of [(2031, 4), 7885, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 2052, F2, 4, 37) (dual of [(2052, 4), 7969, 38]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2238, 2052, F2, 4, 37) (dual of [(2052, 4), 7970, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2238, 8208, F2, 37) (dual of [8208, 7970, 38]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2236, 8206, F2, 37) (dual of [8206, 7970, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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
- 2 times code embedding in larger space [i] based on linear OA(2236, 8206, F2, 37) (dual of [8206, 7970, 38]-code), using
- OOA 4-folding [i] based on linear OA(2238, 8208, F2, 37) (dual of [8208, 7970, 38]-code), using
- 21 times duplication [i] based on linear OOA(2238, 2052, F2, 4, 37) (dual of [(2052, 4), 7970, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2239, 2052, F2, 4, 37) (dual of [(2052, 4), 7969, 38]-NRT-code), using
(239−37, 239, 72153)-Net in Base 2 — Upper bound on s
There is no (202, 239, 72154)-net in base 2, because
- 1 times m-reduction [i] would yield (202, 238, 72154)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 441790 579301 179508 580237 433300 702580 722245 555749 587653 083241 008933 548484 > 2238 [i]