Best Known (209−30, 209, s)-Nets in Base 2
(209−30, 209, 624)-Net over F2 — Constructive and digital
Digital (179, 209, 624)-net over F2, using
- t-expansion [i] based on digital (178, 209, 624)-net over F2, using
- 1 times m-reduction [i] based on digital (178, 210, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 35, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 35, 104)-net over F64, using
- 1 times m-reduction [i] based on digital (178, 210, 624)-net over F2, using
(209−30, 209, 2523)-Net over F2 — Digital
Digital (179, 209, 2523)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 2523, F2, 3, 30) (dual of [(2523, 3), 7360, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 2748, F2, 3, 30) (dual of [(2748, 3), 8035, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2209, 8244, F2, 30) (dual of [8244, 8035, 31]-code), using
- strength reduction [i] based on linear OA(2209, 8244, F2, 31) (dual of [8244, 8035, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 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(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(213, 52, F2, 5) (dual of [52, 39, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- strength reduction [i] based on linear OA(2209, 8244, F2, 31) (dual of [8244, 8035, 32]-code), using
- OOA 3-folding [i] based on linear OA(2209, 8244, F2, 30) (dual of [8244, 8035, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 2748, F2, 3, 30) (dual of [(2748, 3), 8035, 31]-NRT-code), using
(209−30, 209, 100466)-Net in Base 2 — Upper bound on s
There is no (179, 209, 100467)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 822 769367 142461 576609 691908 855303 518433 948968 354065 640676 413528 > 2209 [i]