Best Known (180, 180+33, s)-Nets in Base 2
(180, 180+33, 520)-Net over F2 — Constructive and digital
Digital (180, 213, 520)-net over F2, using
- 23 times duplication [i] based on digital (177, 210, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 42, 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, 42, 104)-net over F32, using
(180, 180+33, 1955)-Net over F2 — Digital
Digital (180, 213, 1955)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2213, 1955, F2, 4, 33) (dual of [(1955, 4), 7607, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 2052, F2, 4, 33) (dual of [(2052, 4), 7995, 34]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2212, 2052, F2, 4, 33) (dual of [(2052, 4), 7996, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2212, 8208, F2, 33) (dual of [8208, 7996, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- OOA 4-folding [i] based on linear OA(2212, 8208, F2, 33) (dual of [8208, 7996, 34]-code), using
- 21 times duplication [i] based on linear OOA(2212, 2052, F2, 4, 33) (dual of [(2052, 4), 7996, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 2052, F2, 4, 33) (dual of [(2052, 4), 7995, 34]-NRT-code), using
(180, 180+33, 66226)-Net in Base 2 — Upper bound on s
There is no (180, 213, 66227)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 212, 66227)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6582 738062 436199 774485 010034 563707 332024 277081 261299 167310 698959 > 2212 [i]